Chapter 04.01: Introduction to Matrix Algebra
Learning Objectives
After reading this chapter, you should be able to
1) define what a matrix is.
2) identify special types of matrices, and
3) identify when two matrices are equal.
What does a matrix look like?
Matrices are everywhere. If you have used a spreadsheet such as Excel or written numbers in a table, you have used a matrix. Matrices make presentation of numbers clearer and make calculations easier to program. Look at the matrix below about the sale of tires in a Blowoutr’us store – given by quarter and make of tires.
\[\begin{matrix} Tirestone\\ Michigan\\ Copper\\ \end{matrix} \stackrel{\mbox{Q1. Q2. Q3. Q4}}{\begin{bmatrix} 25 & 20 & 3 & 2 \\ 5 & 10 &15 &25 \\ 6 & 16 &7 & 27 \\ \end{bmatrix}}\]
If one wants to know how many Copper tires were sold in Quarter 4, we go along the row Copper and column Q4 and find that it is 27.
So, what is a matrix?
A matrix is a rectangular array of elements. The elements can be symbolic expressions or/and numbers. Matrix \(\lbrack A\rbrack\) is denoted by
\[\lbrack A\rbrack = \begin{bmatrix} a_{11} & a_{12} & {.......} & a_{1n} \\ a_{21} & a_{22} & {.......} & a_{2n} \\ \vdots & & & \vdots \\ a_{m1} & a_{m2} & {.......} & a_{{mn}} \\ \end{bmatrix}\]
Row \(i\) of \(\lbrack A\rbrack\) has \(n\) elements and is
\[\left\lbrack a_{i1}a_{i2}{....}a_{{in}} \right\rbrack\]
and column \(j\) of \(\lbrack A\rbrack\) has \(m\) elements and is
\[\begin{bmatrix} a_{1j} \\ a_{2j} \\ \vdots \\ a_{{mj}} \\ \end{bmatrix}\]
Each matrix has rows and columns and this defines the size of the matrix. If a matrix \(\lbrack A\rbrack\) has \(m\) rows and \(n\) columns, the size of the matrix is denoted by \(m \times n\). The matrix \(\lbrack A\rbrack\) may also be denoted by \(\lbrack A\rbrack_{m \times n}\) to show that \(\lbrack A\rbrack\) is a matrix with \(m\) rows and \(n\) columns.
Each entry in the matrix is called the entry or element of the matrix and is denoted by \(a_{{ij}}\) where \(i\) is the row number and \(j\) is the column number of the element.
The matrix for the tire sales example could be denoted by the matrix [A] as
\[\ \lbrack A\rbrack = \begin{bmatrix} 25 & 20 & 3 & 2 \\ 5 & 10 & 15 & 25 \\ 6 & 16 & 7 & 27 \\ \end{bmatrix}\]
There are 3 rows and 4 columns, so the size of the matrix is \(3 \times 4\). In the above \(\lbrack A\rbrack\) matrix, \(a_{34} = 27\).
What are the special types of matrices?
Vector: A vector is a matrix that has only one row or one column. There are two types of vectors – row vectors and column vectors.
Row Vector:
If a matrix \(\lbrack B\rbrack\) has one row, it is called a row vector \(\lbrack B\rbrack = \lbrack b_{1}\ b_{2}\ ...\ b_{n}\rbrack\)
and \(n\) is the dimension of the row vector.
Column vector:
If a matrix \(\lbrack C\rbrack\) has one column, it is called a column vector
\[\lbrack C\rbrack = \begin{bmatrix} c_{1} \\ \vdots \\ \vdots \\ c_{m} \\ \end{bmatrix}\]
and \(m\) is the dimension of the vector.
Submatrix: {#submatrix}
If some row(s) or/and column(s) of a matrix \(\lbrack A\rbrack\) are deleted (no rows or columns may be deleted), the remaining matrix is called a submatrix of \(\lbrack A\rbrack\).
Example 3
Find some of the submatrices of the matrix
\[\lbrack A\rbrack = \begin{bmatrix} 4 & 6 & 2 \\ 3 & - 1 & 2 \\ \end{bmatrix}\]
Solution
\[\begin{bmatrix} 4 & 6 & 2 \\ 3 & - 1 & 2 \\ \end{bmatrix},\ \ \begin{bmatrix} 4 & 6 \\ 3 & - 1 \\ \end{bmatrix},\ \ \begin{bmatrix} 4 & 6 & 2 \\ \end{bmatrix},\left\lbrack 4 \right\rbrack,\begin{bmatrix} 2 \\ 2 \\ \end{bmatrix}\]
are some of the submatrices of \(\lbrack A\rbrack\). Can you find other submatrices of \(\lbrack A\rbrack\)?
Square matrix: {#square-matrix}
If the number of rows \(m\) of a matrix is equal to the number of columns \(n\) of a matrix \(\lbrack A\rbrack\), that is, \(m = n\), then \(\lbrack A\rbrack\) is called a square matrix. The entries \(a_{11},a_{22},...,a_{{nn}}\) are called the diagonal elements of a square matrix. Sometimes the diagonal of the matrix is also called the principal or main of the matrix.
Example 4
Give an example of a square matrix.
Solution
\[\lbrack A\rbrack = \begin{bmatrix} 25 & 20 & 3 \\ 5 & 10 & 15 \\ 6 & 15 & 7 \\ \end{bmatrix}\]
is a square matrix as it has the same number of rows and columns, that is, 3. The diagonal elements of \(\lbrack A\rbrack\) are \(a_{11} = 25,\ \ a_{22} = 10,\ \ a_{33} = 7\).
Upper triangular matrix: {#upper-triangular-matrix}
A \(n \times n\) matrix for which \(a_{{ij}} = 0,\ \ i > j\) for all \(i,j\) is called an upper triangular matrix. That is, all the elements below the diagonal entries are zero.
Lower triangular matrix: {#lower-triangular-matrix}
A \(n \times n\) matrix for which \(a_{{ij}} = 0,\ \ j > i\) for all \(i,j\) is called a lower triangular matrix. That is, all the elements above the diagonal entries are zero.
Diagonal matrix: {#diagonal-matrix}
A square matrix with all non-diagonal elements equal to zero is called a diagonal matrix, that is, only the diagonal entries of the square matrix can be non-zero, (\(a_{{ij}} = 0,\ \ i \neq j\)).
Example 7
Give examples of a diagonal matrix.
Solution
\[\lbrack A\rbrack = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2.1 & 0 \\ 0 & 0 & 5 \\ \end{bmatrix}\]
is a diagonal matrix.
Any or all the diagonal entries of a diagonal matrix can be zero. For example
\[\lbrack A\rbrack = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2.1 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}\]
is also a diagonal matrix.
Identity matrix: {#identity-matrix}
A diagonal matrix with all diagonal elements equal to 1 is called an identity matrix, (\(a_{{ij}} = 0,\ \ i \neq j\) for all \(i,j\) and \(a_{{ii}} = 1\) for all \(i\)).
Zero matrix:
A matrix whose all entries are zero is called a zero matrix, (\(a_{{ij}} = 0\) for all \(i\) and \(j\)).
Example 9
Give examples of a zero matrix.
Solution
\[\lbrack A\rbrack = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}\]
\[\lbrack B\rbrack = \begin{bmatrix} 0 & 0 & 0 \\ 0&0&0 \\ \end{bmatrix}\]
\[\lbrack C\rbrack = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} \]
\[\lbrack D\rbrack = \begin{bmatrix} 0 & 0 & 0 \\ \end{bmatrix}\]
are all examples of a zero matrix.
Tridiagonal matrices:
A tridiagonal matrix is a square matrix in which all elements not on the following are zero - the major diagonal, the diagonal above the major diagonal, and the diagonal below the major diagonal.
Do non-square matrices have diagonal entries?
Yes, for a \(m \times n\) matrix \(\lbrack A\rbrack\) , the diagonal entries are \(a_{11},a_{22}...,a_{k - 1,k - 1},a_{{kk}}\) where \(k = min\{ m,\ n\}\).
Diagonally Dominant Matrix: {#diagonally-dominant-matrix}
A \(n \times n\) square matrix \(\lbrack A\rbrack\) is a diagonally dominant matrix (also called weak diagonally dominant) if
\[\left| a_{{ii}} \right| \geq \sum_{\begin{matrix} j = 1 \\ i \neq j \\ \end{matrix}}^{n}{|a_{{ij}}|}\]
$$$$ for \(i = 1,2,.....,n\)
that is, for each row, the absolute value of the diagonal element is greater than or equal to the sum of the absolute values of the rest of the elements of that row.
Example 12
Give examples of diagonally dominant matrices and not diagonally dominant matrices.
Solution
The matrix \[\lbrack A\rbrack = \begin{bmatrix} 15 & 6 & 7 \\ 2 & - 4 & - 2 \\ 3 & 2 & 6 \\ \end{bmatrix}\]
is a diagonally dominant matrix as
\[\left| a_{11} \right| = \left| 15 \right| = 15 \geq \left| a_{12} \right| + \left| a_{13} \right| = \left| 6 \right| + \left| 7 \right| = 13\]
\[\left| a_{22} \right| = \left| - 4 \right| = 4 \geq \left| a_{21} \right| + \left| a_{23} \right| = \left| 2 \right| + \left| - 2 \right| = 4\]
\[\left| a_{33} \right| = \left| 6 \right| = 6 \geq \left| a_{31} \right| + \left| a_{32} \right| = \left| 3 \right| + \left| 2 \right| = 5\]
The matrix
\[\lbrack B\rbrack = \begin{bmatrix} - 15 & 6 & 9 \\ 2 & - 4 & 2 \\ 3 & - 2 & 5 \\ \end{bmatrix}\]
is a diagonally dominant matrix as
\[\left| b_{11} \right| = \left| - 15 \right| = 15 \geq \left| b_{12} \right| + \left| b_{13} \right| = \left| 6 \right| + \left| 9 \right| = 15\]
\[\left| b_{22} \right| = \left| - 4 \right| = 4 \geq \left| b_{21} \right| + \left| b_{23} \right| = \left| 2 \right| + \left| 2 \right| = 4\]
\[\left| b_{33} \right| = \left| 5 \right| = 5 \geq \left| b_{31} \right| + \left| b_{32} \right| = \left| 3 \right| + \left| - 2 \right| = 5\]
The inequalities are satisfied for all rows.
The matrix
\[\lbrack C\rbrack = \begin{bmatrix} - 15.01 & 6 & 9 \\ 2 & - 4 & 2 \\ 3 & - 2 & 5 \\ \end{bmatrix}\]
is a diagonally dominant matrix as
\[\left| c_{11} \right| = \left| - 15.01 \right| = 15.01 \geq \left| c_{12} \right| + \left| c_{13} \right| = \left| 6 \right| + \left| 9 \right| = 15\]
\[\left| c_{22} \right| = \left| - 4 \right| = 4 \geq \left| c_{21} \right| + \left| c_{23} \right| = \left| 2 \right| + \left| 2 \right| = 4\]
\[\left| c_{33} \right| = \left| 5 \right| = 5 \geq \left| c_{31} \right| + \left| c_{32} \right| = \left| 3 \right| + \left| - 2 \right| = 5\]
The inequalities are satisfied for all rows.
The matrix
\[\left\lbrack D \right\rbrack = \begin{bmatrix} 25 & 5 & 1 \\ 64 & 8 & 1 \\ 144 & 12 & 1 \\ \end{bmatrix}\]
is not diagonally dominant as
\[\left| d_{22} \right| = \left| 8 \right| = 8 \leq \left| d_{21} \right| + \left| d_{23} \right| = \left| 64 \right| + \left| 1 \right| = 65\]
The inequalities are not satisfied for all rows.
When are two matrices considered to be equal?
Two matrices [A] and [B] are equal if the size of [A] and [B] is the same (number of rows and columns of [A] are same as that of [B]) and \(a_{{ij}} = b_{{ij}}\) for all i and j.
Example 13
What would make
\[\lbrack A\rbrack = \begin{bmatrix} 2 & 3 \\ 6 & 7 \\ \end{bmatrix}\]
to be equal to
\[\lbrack B\rbrack = \begin{bmatrix} b_{11} & 3 \\ 6 & b_{22} \\ \end{bmatrix}\]
Solution
The two matrices \(\lbrack A\rbrack\)and \(\lbrack B\rbrack\) would be equal if \(b_{11} = 2\) and \(b_{22} = 7\).
Appendix A
In this appendix, we go beyond the discussion of a diagonally dominant matrix, and discuss weak diagonally dominant matrix, strictly diagonally dominant matrix, irreducible diagonally dominant matrix and reducible matrices. We also discuss the implications of such matrices.
What is a diagonally dominant matrix?
A \(n \times n\) square matrix \([A]\) is a diagonally dominant matrix if
\[\left|a_{ii}\right| \geq \sum^n_{\substack{j=1\\ i \neq j}} \left|a_{ij}\right| \text{ for } i=1,2,....,n\]
that is, for each row, the absolute value (also called magnitude) of the diagonal element is greater than or equal to the sum of the absolute values of the rest of the elements of that row.
Example A.1
Give examples of matrices that are diagonally dominant and those that are not diagonally dominant.
Solution
The matrix
\[[A]=\begin{bmatrix}15 & 6 & 7\\ 2 & -4.1 & -2 \\ 3 & 2 & 6\end{bmatrix}\]
is a diagonally dominant matrix.
Why? Because for each and every row, the answer to the question below is Yes.
Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\) ? Yes, because
\[\left|a_{11}\right|=|15|=15,\left|a_{12}\right|+\left|a_{13}\right|=|6|+|7|=13,15 \geq 13\]
Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\) ? Yes, because
\[\left|a_{22}\right|=|-4.1|=4.1,\left|a_{21}\right|+\left|a_{23}\right|=|2|+|-2|=4,4.1 \geq 4\]
Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\) ? Yes, because
\[\left|a_{33}\right|=|6|=6,\left|a_{31}\right|+\left|a_{32}\right|=|3|+|2|=5,6 \geq 5\]
____________________________________
The matrix
\[[B] = \begin{bmatrix} -15 & 6 & 9 \\ 2 & -4 & -2 \\ 3 & -2 & 5 \end{bmatrix}\]
is a diagonally dominant matrix.
Why? Because for each and every row, the answer to the question below is Yes.
Row 1: Is \(\left|b_{11}\right| \geq\left|b_{12}\right|+\left|b_{13}\right|\) ? Yes, because
\[\left|b_{11}\right|=|15|=15,\left|b_{12}\right|+\left|b_{13}\right|=|6|+|9|=15,15 \geq 15\]
Row 2: Is \(\left|b_{22}\right| \geq\left|b_{21}\right|+\left|b_{23}\right|\) ? Yes, because
\[\left|b_{22}\right|=|-4|=4, \quad\left|b_{21}\right|+\left|b_{23}\right|=|2|+|-2|=4,4 \geq 4\]
Row 3: Is \(\left|b_{33}\right| \geq\left|b_{31}\right|+\left|b_{32}\right|\)? Yes, because
\[\left|b_{33}\right|=|5|=5,\left|b_{31}\right|+\left|b_{32}\right|=|3|+|2|=5,5 \geq 5\]
________________________________________
The matrix
\[[C] = \begin{bmatrix} -15 & 6 & 9 \\ 2 & -4.1 & -2 \\ 3 & -2 & 5 \end{bmatrix}\]
is a diagonally dominant matrix.
Why? Because for each and every row, the answer to the question below is Yes.
Row 1: Is \(\left|c_{11}\right| \geq\left|c_{12}\right|+\left|c_{13}\right|\)? Yes, because
\[\left|c_{11}\right|=|15|=15,\left|c_{12}\right|+\left|c_{13}\right|=|6|+|9|=15,15 \geq 15\]
Row 2: Is \(\left|c_{22}\right| \geq\left|c_{21}\right|+\left|c_{23}\right|\)? Yes, because
\[\left|c_{22}\right|=|-4.1|=4,\left|c_{21}\right|+\left|c_{23}\right|=|2|+|-2|=4,4.1 \geq 4\]
Row 3: Is \(\left|c_{33}\right| \geq\left|c_{31}\right|+\left|c_{32}\right|\)? Yes, because
\[\left|c_{33}\right|=|5|,\left|c_{31}\right|+\left|c_{32}\right|=|3|+|2|=5,5 \geq 5\]
________________________________________
The matrix
\[[D] = \begin{bmatrix} 25 & 5 & 1 \\ 64 & 8 & 1 \\ 144 & 12 & 1 \end{bmatrix}\]
is not a diagonally dominant matrix.
Why? Because for each and every row, the answer to the question below is not a Yes.
Row 1: Is \(\left|d_{11}\right| \geq\left|d_{12}\right|+\left|d_{13}\right|\)? Yes, because
\[\left|d_{11}\right|=\left|25\right|=25 ,\left|d_{12}\right|+\left|d_{13}\right|=|5|+|1|=6,25 \geq 6\]
Row 2: Is \(\left|d_{22}\right| \geq\left|d_{21}\right|+\left|d_{23}\right|\)? No, because
\[\left|d_{22}\right|=|8|=8, \left|d_{21}\right|+\left|d_{23}\right|=|64|+|1|=65,8<65\]
Row 3: Is \(\left|d_{33}\right| \geq\left|d_{31}\right|+\left|d_{32}\right|\)? No, because
\[\left|d_{33}\right|=|1|=1,\left|d_{31}\right|+\left|d_{32}\right|=|144|+|12|=156, 1<156\]
What is a weak diagonally dominant matrix?
The answer is simple – the definition of a weak(ly) diagonally dominant matrix is identical to that of a diagonally dominant matrix as the inequality used for the check is a weak inequality of greater than or equal to (\(\geq\)).
What is a strictly diagonally dominant matrix?
A \(n \times n\) square matrix is a strictly diagonally dominant matrix if
\[\left|a_{ii}\right| > \sum^n_{\substack{j=1\\ i \neq j}} \left|a_{ij}\right| \text{ for } i=1,2,....,n\]
that is, for each row, the absolute value of the diagonal element is strictly greater than the sum of the absolute values of the rest of the elements of that row.
Example A.2
Give examples of strictly diagonally dominant matrices and not strictly diagonally dominant matrices.
Solution
The matrix
\[[A] = \begin{bmatrix} 15 & 6 & 7 \\ 2 & -4.1 & -2 \\ 3 & 2 & 6 \end{bmatrix}\]
is a strictly diagonally dominant matrix
Why? Because for each and every row, the answer to the question below is Yes.
Row 1: Is \(\left|a_{11}\right| >\left|a_{12}\right|+\left|a_{13}\right|\)? Yes, because
\[\left|a_{11}\right|=|15|=15,\left|a_{12}\right|+\left|a_{13}\right|=|6|+|7|=13,15>13 .\]
Row 2: Is \(\left|a_{22}\right| >\left|a_{21}\right|+\left|a_{23}\right|\)? Yes, because
\[\left|a_{22}\right|=|-4.1|=4.1,\left|a_{21}\right|+\left|a_{23}\right|=|2|+|-2|=4,4.1>4\]
Row 3: Is \(\left|a_{33}\right| > \left|a_{31}\right|+\left|a_{32}\right|\)? Yes, because
\[\left|a_{33}\right|=|6|=6,\left|a_{31}\right|+\left|a_{32}\right|=|3|+|2|=5,6>5\] ________________________________________
The matrix
\[[B]=\begin{bmatrix} 13 & 6 & 7 \\ 2 & -4.1 & -2 \\ 3 & 2 & 6 \end{bmatrix}\]
is a not a strictly diagonally dominant matrix
Why? Because for each and every row, the answer to the question below is not a Yes.
Row 1: Is \(\left|b_{11}\right| > \left|b_{12}\right|+\left|b_{13}\right|\)? No, because
\[\left|b_{11}\right|=|13|=13,\left|b_{12}\right|+\left|b_{13}\right|=|6|+|7|=13,13 = 13\]
Row 2: Is \(\left|b_{22}\right| >\left|b_{21}\right|+\left|b_{23}\right|\)? Yes, because
\[\left|b_{22}\right|=|-4.1|=4.1,\left|b_{21}\right|+\left|b_{23}\right|=|2|+|-2|=4,4.1>4\]
Row 3: Is \(\left|b_{33}\right| >\left|b_{31}\right|+\left|b_{32}\right|\)? Yes, because
\[\left|b_{33}\right|=|6|=6,\left|b_{31}\right|+\left|b_{32}\right|=|3|+|2|=5,6>5\]
________________________________________
The matrix
\[[C]=\begin{bmatrix} 25 & 5 & 1 \\ 64 & 8 & 1 \\ 144 & 12 & 1 \end{bmatrix}\]
is not a strictly diagonally dominant matrix.
Why? Because for each and every row, the answer to the question below is not a Yes.
Row 1: Is \(\left|c_{11}\right| \geq\left|c_{12}\right|+\left|c_{13}\right|\)? Yes, because
\[\left|c_{11}\right|=|25|=25,\left|c_{12}\right|+\left|c_{13}\right|=|5|+|1|=6,25>6\]
Row 2: Is \(\left|c_{22}\right| \geq\left|c_{21}\right|+\left|c_{23}\right|\)? No, because
\[\left|c_{22}\right|=|8|=8, \left|c_{21}\right|+\left|c_{23}\right|=|64|+|1|=65,8<65\]
Row 3: Is \(\left|c_{33}\right| \geq\left|c_{31}\right|+\left|c_{32}\right|\)? No, because
\[\left|c_{33}\right|=|1|=1, c_{31}|+| c_{32}|=| 144|+| 12 \mid=156, 1<156\]
What is an irreducible diagonally dominant matrix?
A \(n \times n\) square matrix is an irreducible diagonally dominant matrix if
\[[A]\ \text{is irreducible},\]
\[\left|a_{ii}\right| \geq \sum^n_{\substack{j=1\\ i \neq j}} \left|a_{ij}\right| \text{ for } i=1,2,....,n\ \text{and}\]
\[\left|a_{ii}\right| > \sum^n_{\substack{j=1\\ i \neq j}} \left|a_{ij}\right| \text{ for at least one row, } i=1,2,....,n\]
The second condition means that for each row, the absolute value (also called magnitude) of the diagonal element is greater than or equal to the sum of the absolute values of the rest of the elements of that row. The third condition means that for at least one row, the absolute value (also called magnitude) of the diagonal element is greater than the sum of the absolute values of the rest of the elements of that row.
Example
Give examples of matrices that are irreducibly diagonally dominant and those that are not irreducibly diagonally dominant.
Solution
The matrix
\[[A]=\begin{bmatrix} 15 & 6 & 7 \\ 2 & -4.1 & -2 \\ 3 & 2 & 6 \end{bmatrix}\]
is an irreducible diagonally dominant matrix.
Because \([A]\) irreducible and the answer to every question below is Yes.
Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\)? Yes, because
\[\left|a_{11}\right|=|15|,\left|a_{12}\right|+\left|a_{13}\right|=|6|+|7|=13,15>13 .\]
Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\)? Yes, because
\[\left|a_{22}\right|=|-4.1|=4.1,\left|a_{21}\right|+\left|a_{23}\right|=|2|+|-2|=4,4.1>4\]
Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\)? Yes, because
\[\left|a_{33}\right|=|6|\left|a_{31}\right|+\left|a_{32}\right|=|3|+|2|=5,6>5\]
Is the inequality satisfied strictly for at least one row? Yes, it is satisfied for Row 1 (Row 2 and Row 3 do too satisfy the inequality strictly but is not necessary)
________________________________________
The matrix
\[[B]=\begin{bmatrix} -15 & 6 & 9 \\ 2 & -4 & -2 \\ 3 & -2 & 5 \end{bmatrix}\]
is a not an irreducible diagonally dominant matrix.
Because \([B]\) irreducible but the answer to every question below is not a Yes.
Row 1: Is \(\left|b_{11}\right| \geq\left|b_{12}\right|+\left|b_{13}\right|\)? Yes, because
\[\left|b_{11}\right|=|15|=15,\left|b_{12}\right|+\left|b_{13}\right|=|6|+|9|=15,15 \geq 15\]
Row 2: Is \(\left|b_{22}\right| \geq\left|b_{21}\right|+\left|b_{23}\right|\)? Yes, because
\[\left|b_{22}\right|=|-4|=4, \left|b_{21}\right|+\left|b_{23}\right|=|2|+|-2|=4,4 \geq 4\]
Row 3: Is \(\left|b_{33}\right| \geq\left|b_{31}\right|+\left|b_{32}\right|\)? Yes, because
\[\left|b_{33}\right|=|5|=5,\left|b_{31}\right|+\left|b_{32}\right|=|3|+|2|=5,5 \geq 5\]
Is the inequality satisfied strictly for at least one row? No.
________________________________________
The matrix
\[[C]=\begin{bmatrix} -15 & 6 & 9 \\ 2 & -4.1 & -2 \\ 3 & -2 & 5 \end{bmatrix}\]
is an irreducible diagonally dominant matrix.
Because \([C]\) irreducible and the answer to every question below is Yes.
Row 1: Is \(\left|c_{11}\right| \geq\left|c_{12}\right|+\left|c_{13}\right|\)? Yes, because
\[\left|c_{11}\right|=|15|,\left|c_{12}\right|+\left|c_{13}\right|=|6|+|9|=15,15 \geq 15\]
Row 2: Is \(\left|c_{22}\right| \geq\left|c_{21}\right|+\left|c_{23}\right|\)? Yes, because
\[\left|c_{22}\right|=|-4.1|=4.1, \left|c_{21}\right|+\left|c_{23}\right|=|2|+|-2|=4,4.1 \geq 4\]
Row 3: Is \(\left|c_{33}\right| \geq\left|c_{31}\right|+\left|c_{32}\right|\)? Yes, because
\[\left|c_{33}\right|=|5|,\left|c_{31}\right|+\left|c_{32}\right|=|3|+|2|=5,5 \geq 5\]
Is the inequality satisfied strictly for at least one row? Yes, it is satisfied for Row 2.
________________________________________
The matrix
\[[D]=\begin{bmatrix} 25 & 5 & 1 \\ 64 & 8 & 1 \\ 144 & 12 & 1 \end{bmatrix}\]
is not an irreducible diagonally dominant matrix.
Because \([D]\) is irreducible but the answer to every question below is not a Yes.
Row 1: Is \(\left|d_{11}\right| \geq\left|d_{12}\right|+\left|d_{13}\right|\)? Yes, because
\[\left|d_{11}\right|=|25|=25,\left|d_{12}\right|+\left|d_{13}\right|=|5|+|1|=6,25>6\]
Row 2: Is \(\left|d_{22}\right| \geq\left|d_{21}\right|+\left|d_{23}\right|\)? No, because
\[\left|d_{22}\right|=|8|=8, \left|d_{21}\right|+\left|d_{23}\right|=|64|+|1|=65,8<65\]
Row 3: Is \(\left|d_{33}\right| \geq\left|d_{31}\right|+\left|d_{32}\right|\)? No, because
\[\left|d_{33}\right|=|1|=1,| d_{31}|+| d_{32}|=| 144|+| 12 \mid=156, 1<156\]
There is no need to check for strict inequality condition.
What is an irreducible matrix?
A square matrix is called reducible matrix if the following is true. Take the indices \(i=1,2,....,n\) and see if they can be divided into two disjoint nonempty sets \(i_1,i_2,....,i_\alpha\) and \(j_1,j_2,....,j_\beta\) such that
\[n=\alpha + \beta,\ \text{and}\]
and
\[a_{{i_k}{j_l}}=0,\ k=1,2,....,\alpha\ \text{and}\ l=1,2,....,\beta\]
If the square matrix is not reducible, it is called an irreducible matrix.
A square matrix \([A]\) is called reducible matrix if and only if for any perturbation matrix \([P]\), the matrix multiplication \([P]^T[A][P]\) results in a block upper triangular matrix.
Give examples of irreducible and reducible matrices.
The matrix
\[[A]=\begin{bmatrix} 0 & 5 & 7 \\ 8 & 0 & 0 \\ 10 & 0 & 0 \end{bmatrix}\]
is an irreducible matrix.
The matrix
\[[B]=\begin{bmatrix} 5 & 0 & 0 \\ 0 & 4 & 6 \\ 10 & 0 & 0 \end{bmatrix}\]
is a reducible matrix. Why? Take the indices \(i=1,2,3\) and see that they can be divided into two disjoint nonempty sets 1 and 2,3 such that,
\[\alpha = 1, \beta = 2,\ \text{giving}\ \alpha + \beta = 1+2 = 3,\ \text{and}\]
\[b_{{i_k}{j_l}}=0,\ k=1 \ \text{and}\ l = 1,2\]
Implications of diagonally dominant matrices?
If a square matrix is strictly diagonally dominant
then the matrix is non-singular [1].
then if the matrix is symmetric with non-negative diagonal entries, the matrix is positive semi-definite [1].
then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Gauss-Seidel numerical method will always converge [2].
then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Jordan numerical method will always converge [2].
then if the diagonal entries of the matrix are positive, the real parts of the matrix eigenvalues are positive [1].
then if the diagonal entries of the matrix are negative, the real parts of the matrix eigenvalues are negative [1].
then if the matrix is column dominant, no pivoting is needed for Gaussian elimination [2].
then if the matrix is column dominant, no pivoting is needed for LU factorization [2].
If a square matrix is irreducible diagonally dominant
then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Gauss-Seidel numerical method will always converge.
then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Jordan numerical method will always converge.
the matrix is non-singular [2].
If a square matrix is diagonally dominant (also called weakly diagonally dominant)
then if the matrix is column dominant, no pivoting is needed for Gaussian elimination [3].
then if the matrix is column dominant, no pivoting is needed for LU factorization [3].
References
[1] Briggs, Keith. “Diagonally Dominant Matrix.” FromMathWorld–A Wolfram
Web Resource, created by Eric W. Weisstein.
http://mathworld.wolfram.com/DiagonallyDominantMatrix.html
[2] Diagonally Dominant Matrix, see
https://en.wikipedia.org/wiki/Diagonally_dominant_matrix, last accessed
on November 4, 2016.
[3] “Lecture 4: A Gaussian Elimination Example”, see
http://www.cs.yale.edu/homes/spielman/BAP/lect4.pdf, last accessed on
November 4, 2016.
Key Terms:
Matrix
Vector
Submatrix
Square matrix
Equal matrices
Zero matrix
Identity matrix
Diagonal matrix
Upper triangular matrix
Lower triangular matrix
Tri-diagonal matrix
Multiple Choice Test
(1). For an \(n \times n\) upper triangular matrix \(\left\lbrack A \right\rbrack\),
(A) \(a_{{ij}} = 0,i > j\)
(B) \(a_{{ij}} = 0,j > i\)
(C) \(a_{{ij}} \neq 0,i > j\)
(D) \(a_{{ij}} \neq 0,j > i\)
(2). Which one of these square matrices is strictly diagonally dominant?
(A) \(\begin{bmatrix} 5 & 7 & 0 \\ 3 & - 6 & 2 \\ 2 & 2 & 9 \\ \end{bmatrix}\)
(B) \(\begin{bmatrix} 7 & - 5 & - 2 \\ 6 & - 13 & - 7 \\ 6 & - 7 & - 13 \\ \end{bmatrix}\)
(C) \(\begin{bmatrix} 8 & - 5 & - 2 \\ 6 & - 14 & - 7 \\ 6 & - 7 & - 13 \\ \end{bmatrix}\)
(D) \(\begin{bmatrix} 8 & 5 & 2 \\ 6 & 14 & 7 \\ 6 & 7.5 & 14 \\ \end{bmatrix}\)
(3). The order of the following matrix is
\[\begin{bmatrix} 4 & - 6 & - 7 & 2 \\ 3 & 2 & - 5 & 6 \\ \end{bmatrix}\]
(A) \(4 \times 2\)
(B) \(2 \times 4\)
(C) \(8 \times 1\)
(D) not defined
(4). To make the following two matrices equal
\[\left\lbrack A \right\rbrack = \begin{bmatrix} 5 & - 6 & 7 \\ 3 & 2 & 5 \\ \end{bmatrix}\]
\[\left\lbrack B \right\rbrack = \begin{bmatrix} 5 & p & 7 \\ 3 & 2 & 5 \\ \end{bmatrix}\] the value of \(p\) is
(A) \(- 6\)
(B) \(6\)
(C) \(0\)
(D) \(7\)
(5). For a square \(n \times n\) matrix \(\left\lbrack A \right\rbrack\) to be an identity matrix,
(A) \(a_{{ij}} \neq 0,i = j;a_{{ij}} = 0,i = j\)
(B) \(a_{{ij}} = 0,i \neq j;a_{{ij}} = 1,i = j\)
(C) \(a_{{ij}} = 0,i \neq j;a_{{ij}} = i,i = j\)
(D) \(a_{{ij}} = 0,i \neq j;a_{{ij}} > 0,i = j\)
(6). To make the following square matrix to be diagonally dominant, the value of \(p\) needs to be
\[\begin{bmatrix} 6 & - 2 & - 4 \\ 7 & 9 & 1 \\ 8 & - 5 & p \\ \end{bmatrix}\]
(A) greater than or equal to 13
(B) greater than 3
(C) greater than or equal to 3
(D) greater than 13
For complete solution, go to
https://ma.mathforcollege.com/mcquizzes/04sle/quiz_04sle_introduction_solution.pdf
Multiple Choice Test
(1). Given
\[[A] =\begin{bmatrix} 6 & 2 & 3 & 9 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 4 & 5 \\ 0 & 0 & 0 & 6 \\ \end{bmatrix}\]
then \([A]\) is a ______________ matrix.
(A) diagonal
(B) identity
(C) lower triangular
(D) upper triangular
(2). A square matrix \([A]\) is lower triangular if
(A) \(a_{{ij}} = 0,j > i\)
(B) \(a_{{ij}} = 0,i > j\)
(C) \(a_{{ij}} \neq 0,i > j\)
(D) \(a_{{ij}} \neq 0,j > i\)
(3). Given
\[\lbrack A\rbrack = \begin{bmatrix} 12.3 & - 12.3 & 20.3 \\ 11.3 & - 10.3 & - 11.3 \\ 10.3 & - 11.3 & - 12.3 \\ \end{bmatrix},\ \lbrack B\rbrack = \begin{bmatrix} 2 & 4 \\ - 5 & 6 \\ 11 & - 20 \\ \end{bmatrix}\]
then if
\([C] = [A] [B]\), then
\(c_{31}=\) _____________________
(A) \(-58.2\)
(B) \(-37.6\)
(C) \(219.4\)
(D) \(259.4\)
(4). The following system of equations has ____________ solution(s).
\[x + y = 2\]
\[6x+6y =12\]
(A) infinite
(B) no
(C) two
(D) unique
(5). Consider there are only two computer companies in a country. The companies are named Dude and Imac. Each year, company Dude keeps 1/5th of its customers, while the rest switch to Imac. Each year, Imac keeps 1/3rd of its customers, while the rest switch to Dude. If in 2003, Dude had 1/6th of the market and Imac had 5/6th of the market, what will be the share of Dude computers when the market becomes stable?
(A) \(37/90\)
(B) \(5/11\)
(C) \(6/11\)
(D) \(53/90\)
(6). Three kids - Jim, Corey, and David receive an inheritance of \(\$2,253,453\). The money is put in three trusts but is not divided equally, to begin with. Corey’s trust is three times that of David’s because Corey made an A in Dr. Kaw’s class. Each trust is put in an interest-generating investment. The three trusts of Jim, Corey, and David pays an interest of \(6\%\), \(8\%\), \(11\%\), respectively. The total interest of all the three trusts combined at the end of the first year is \(\$190,740.57\). The equations to find the trust money of Jim (\(J\)), Corey (\(C\)), and David (\(D\)) in a matrix form is
(A) \(\begin{bmatrix} 1 & 1 & 1 \\ 0 & 3 & - 1 \\ 0.06 & 0.08 & 0.11 \\ \end{bmatrix}\begin{bmatrix} J \\ C \\ D \\ \end{bmatrix} = \begin{bmatrix} 2,253,453 \\ 0 \\ 190,740.57 \\ \end{bmatrix}\)
(B) \(\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & - 3 \\ 0.06 & 0.08 & 0.11 \\ \end{bmatrix}\begin{bmatrix} J \\ C \\ D \\\end{bmatrix} = \begin{bmatrix} 2,253,453 \\ 0 \\ 190,740.57 \\ \end{bmatrix}\)
(C) \(\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & - 3 \\ 6 & 8 & 11 \\ \end{bmatrix}\begin{bmatrix} J \\ C \\ D \\ \end{bmatrix} = \begin{bmatrix} 2,253,453 \\ 0 \\ 190,740.57 \\ \end{bmatrix}\)
(D) \(\begin{bmatrix} 1 & 1 & 1 \\ 0 & 3 & - 1 \\ 6 & 8 & 11 \\ \end{bmatrix}\begin{bmatrix} J \\ C \\ D \\ \end{bmatrix} = \begin{bmatrix} 2,253,453 \\ 0 \\ 19,074,057 \\ \end{bmatrix}\)
For complete solution, go to
http://nm.mathforcollege.com/mcquizzes/04sle/quiz_04sle_background_solution.pdf
Introduction Exercise
(1). Write an example of a row vector of dimension 4.
Answer: \(\begin{bmatrix} 5 & 6 & 2 & 3 \\ \end{bmatrix}\)
(2). Write an example of a column vector of dimension 4.
Answer:\(\ \begin{bmatrix} 5 \\ - 7 \\ 3 \\ 2.5 \\ \end{bmatrix}\)
(3). Write an example of a square matrix of order \(4 \times 4\).
Answer:\(\ \begin{bmatrix} 9 & 0 & - 2 & 3 \\ - 2 & 3 & 5 & 1 \\ 1.5 & 6 & 7 & 8 \\ 1.1 & 2 & 3 & 4 \\ \end{bmatrix}\)
(4). Write an example of a tri-diagonal matrix of order \(4 \times 4\).
Answer:\(\ \begin{bmatrix} 6 & 3 & 0 & 0 \\ 2.1 & 2 & 2.2 & 0 \\ 0 & 6.2 & - 3 & 3.5 \\ 0 & 0 & 2.1 & 4.1 \\ \end{bmatrix}\)
(5). Write an example of a identity matrix of order \(5 \times 5\).
Answer:\(\ \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}\)
(6). Write an example of a upper triangular matrix of order \(4 \times 4\).
Answer:\(\ \begin{bmatrix} 6 & 2 & 3 & 9 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 4 & 5 \\ 0 & 0 & 0 & 6 \\ \end{bmatrix}\)
(7). Write an example of a lower triangular matrix of order \(4 \times 4\).
Answer:\(\ \begin{bmatrix} 2 & 0 & 0 & 0 \\ 3 & 1 & 0 & 0 \\ 4 & 2 & 4 & 0 \\ 5 & 3 & 5 & 6 \\ \end{bmatrix}\)
(8). Which of these matrices are strictly diagonally dominant?
\(\left\lbrack A \right\rbrack = \begin{bmatrix} 15 & 6 & 7 \\ 2 & - 4 & 2 \\ 3 & 2 & 6 \\ \end{bmatrix}\)
\(\left\lbrack A \right\rbrack = \begin{bmatrix} 5 & 6 & 7 \\ 2 & - 4 & 2 \\ 3 & 2 & - 5 \\ \end{bmatrix}\)
\(\left\lbrack A \right\rbrack = \begin{bmatrix} 5 & 3 & 2 \\ 6 & - 8 & 2 \\ 7 & - 5 & 12 \\ \end{bmatrix}\)
Answer: (A) Yes (B) No (C) No
(9). Find all the submatrices of
\[\lbrack A\rbrack = \begin{bmatrix} 10 & - 7 & 0 \\ 0 & - 0.001 & 6 \\ \end{bmatrix}\]
Answer: \(\left\lbrack 10 \right\rbrack\) \(\left\lbrack - 7 \right\rbrack\) , \(\left\lbrack 0 \right\rbrack\), \(\left\lbrack - 0.001 \right\rbrack\), \(\left\lbrack 6 \right\rbrack\) \(\begin{bmatrix} 10 \\ 0 \\ \end{bmatrix}\), \(\begin{bmatrix} - 7 \\ - .001 \\ \end{bmatrix}\), \(\begin{bmatrix} 0 \\ 6 \\ \end{bmatrix}\), \(\begin{bmatrix} 10 & - 7 & 0 \\ \end{bmatrix}\), \(\begin{bmatrix} 0 & - 0.001 & 6 \\ \end{bmatrix}\), \(\begin{bmatrix} 10 & - 7 \\ 0 & - 0.001 \\ \end{bmatrix}\),\(\begin{bmatrix} 10 & 0 \\ 0 & 6 \\ \end{bmatrix}\), \(\begin{bmatrix} - 7 & 0 \\ - 0.001 & 6 \\ \end{bmatrix}\), \(\left\lbrack 10, - 7 \right\rbrack\) , \(\left\lbrack 10,0 \right\rbrack\), \(\left\lbrack - 7,0 \right\rbrack\) , \(\left\lbrack 0,6 \right\rbrack\) , \(\left\lbrack 0, - 0.001 \right\rbrack\), \(\left\lbrack - 0.001,6 \right\rbrack\).
(10). If
\[\lbrack A\rbrack = \begin{bmatrix} 4 & - 1 \\ 0 & 2 \\ \end{bmatrix},\]
what are \(b_{11}\) and \(b_{12}\) in
\[\lbrack B\rbrack = \begin{bmatrix} b_{11} & b_{12} \\ 0 & 4 \\ \end{bmatrix}\]
if \(\lbrack B\rbrack = 2\lbrack A\rbrack\).
Answer:\(\ 8, - 2\)
(11). Are matrix
\[\lbrack A\rbrack = \begin{bmatrix} 10 & - 7 & 0 \\ 0 & - 0.001 & 6 \\ \end{bmatrix}\]
and matrix
\[\lbrack B\rbrack = \begin{bmatrix} 10 & 0 \\ - 7 & - 0.001 \\ 0 & 6 \\ \end{bmatrix}\]
equal?
Answer: No
(12). A square matrix \(\lbrack A\rbrack\) is lower triangular if
\(a_{{ij}} = 0\) for \(i > j\)
\(a_{{ij}} = 0\) for \(j > i\)
\(a_{{ij}} = 0\) for \(i = j\)
\(a_{{ij}} = 0\) for \(i + j = odd\ integer\)
Answer: B
(13). A square matrix \(\lbrack A\rbrack\) is upper triangular if
\(a_{{ij}} = 0\) for \(i > j\)
\(a_{{ij}} = 0\) for \(j > i\)
\(a_{{ij}} = 0\) for \(i = j\)
\(a_{{ij}} = 0\) for \(i + j = odd\ integer\)
Answer: A
Problem Set
(1). Write an example of a row vector of dimension \(4\).
Answer: \(\left[B\right]=\left[\begin{matrix}2&1&5&7\\\end{matrix}\right]\)
(2). Write an example of a column vector of dimension \(4\).
Answer: \(\left[A\right]=\left[\begin{matrix}1\\8\\9\\5\\\end{matrix}\right]\)
(3). Write an example of a square matrix of order \(4 \times 4\).
Answer: \(\left[A\right]=\left[\begin{matrix}1&8&6&2\\9&3&4&1\\7&6&2&5\\8&3&6&4\\\end{matrix}\right]\)
(4). Write an example of a tri-diagonal matrix of order \(4 \times 4\).
Answer: \(\left[A\right]=\left[\begin{matrix}2&4&0&0\\2&3&9&0\\0&0&5&2\\0&0&3&6\\\end{matrix}\right]\)
(5). Write an example of a identity matrix of order \(5 \times 5\).
Answer: \(\left[A\right]=\left[\begin{matrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\\end{matrix}\right]\)
(6). Write an example of a upper triangular matrix of order \(4 \times 4\).
Answer: \(\left[A\right]=\left[\begin{matrix}1&3&6&0\\0&5&8&2\\0&0&4&9\\0&0&0&2\\\end{matrix}\right]\)
(7). Write an example of a lower triangular matrix of order \(4 \times 4\).
Answer: \(\left[A\right]=\left[\begin{matrix}1&0&0&0\\3&5&0&0\\9&6&4&0\\7&8&1&2\\\end{matrix}\right]\)
(8). Which of these matrices are strictly diagonally dominant?
a) \[[A]= \begin{bmatrix} 15 & 6 & 7 \\ 2 & - 4 & 2 \\ 3 & 2 & 6 \\ \end{bmatrix}\]
b)
\[[A]= \begin{bmatrix}
5 & 6 & 7 \\
2 & - 4 & 2 \\
3 & 2 & - 5 \\
\end{bmatrix}\]
c)
\[[A]= \begin{bmatrix}
5 & 3 & 2 \\
6 & - 8 & 2 \\
7 & - 5 & 12 \\
\end{bmatrix}\]
Answer: None of them
(9). For the following matrices
\[[A] = \begin{bmatrix} 3 & 0 \\ -1 & 2 \\ 1 & 1 \end{bmatrix},\ [B] = \begin{bmatrix} 4 & - 1 \\ 0 & 2 \\ \end{bmatrix},\ [C] =\begin{bmatrix} 5 & 2\\ 3 & 5\\ 6 & 7 \end{bmatrix},\]
find where possible
a) 4[A]+5[C]
b) [A][B]
c) [A]-2[C]
Answer: a) \(\left[C\right]=\left[\begin{matrix}37&10\\11&33\\34&39\\\end{matrix}\right]\)
b) \(\left[C\right]=\left[\begin{matrix}12&-3\\-4&5\\4&1\\\end{matrix}\right]\)
c) \(\left[C\right]=\left[\begin{matrix}-7&-4\\-7&-8\\-11&-13\\\end{matrix}\right]\)
(10). Food orders are taken from two engineering departments for a takeout. The order is tabulated below.
Food order:
\[\begin{matrix} \text{Mechanical} \\ \text{Civil} \\ \end{matrix}\overset{\begin{matrix} \begin{matrix} \text{Chicken } \\ \ \text{Sandwich} \\ \end{matrix} & \text{Fries} & \text{Drink} \\ \end{matrix}}{\begin{bmatrix} 25 & 35 & 25 \\ 21 & 20 & 21 \\ \end{bmatrix}}\]
However they have a choice of buying this food from three different restaurants. Their prices for the three food items are tabulated below
Price Matrix:
\[\begin{matrix} \text{Chicken}\text{Sandwich} \\ \text{Fries} \\ \text{Drink} \\ \end{matrix}\overset{\begin{matrix} \text{McFat} & \text{Burcholestrol} & \begin{matrix} \ \text{Kentucky} \\ \ \text{Sodium} \\ \end{matrix} \\ \end{matrix}}{\begin{bmatrix} 2.42\quad & 2.38 & 2.46 \\ 0.93\quad & 0.90 & 0.89 \\ 0.95\quad & 1.03 & 1.13 \\ \end{bmatrix}}\]
Show how much each department will pay for their order at each restaurant. Which restaurant would be more economical to order from for each department?
Answer: Burcholesterol is the cheapest for the mechanical department, and McFat is the cheapest for the civil department.
(11). Given
\[[A]= \begin{bmatrix} 12.3 & - 12.3 & 10.3 \\ 11.3 & - 10.3 & - 11.3 \\ 10.3 & - 11.3 & - 12.3 \\ \end{bmatrix},\]
\[[B] = \begin{bmatrix} 2 & 4 \\ - 5 & 6 \\ 11 & - 20 \\ \end{bmatrix}\]
Find \([C]\) if \([C] = [A] [B]\) .
Answer: \(\left[C\right]=\left[\begin{matrix}199.4&-230.6\\-50.2&209.4\\-58.2&219.4\\\end{matrix}\right]\)
(12). The set of equations
\[\begin{bmatrix} 1 & 2 & 5 \\ 7 & 3 & 9 \\ 8 & 5 & 14 \\ \end{bmatrix}\ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} = \begin{bmatrix} 8 \\ 19 \\ 27 \\ \end{bmatrix}\]
has unique, no, or infinite solutions. Which is it?
Answer: infinite solutions
(13). For what values of \(a\) will the following simultaneous linear equations
\[x_{1} + x_{2} + x_{3} = 4\] \[x_{3} = 2\] \[\left( a^{2} - 4 \right)x_{1} + x_{3} = a - 2\] have
a) Unique solution
b) No solution
c) Infinite solutions
Answer: Intentionally not provided
(14). Find if
\[[A]= \begin{bmatrix} 5 & - 2.5 \\ - 2 & 3 \\ \end{bmatrix}\]
and
\[[B]= \begin{bmatrix} 0.3 & 0.25 \\ 0.2 & 0.5 \\ \end{bmatrix}\]
are inverse of each other.
Answer: Yes
(15). Find if
\[[A]= \begin{bmatrix} 5 & 2.5 \\ 2 & 3 \\ \end{bmatrix}\]
and
\[[B]= \begin{bmatrix} 0.3 & - 0.25 \\ 0.2 & 0.5 \\ \end{bmatrix}\]
are inverse of each other.
Answer: No
(16). Three kids - Jim, Corey and David receive an inheritance of \(\$2,253,453\). The money is put in three trusts but is not divided equally to begin with. Corey gets three times what David gets because Corey made an “A” in Dr. Kaw’s class. Each trust is put in an interest generating investment. The three trusts of Jim, Corey and David pays an interest of \(6\%\), \(8\%\), \(11\%\), respectively. The combined interest gained in all the three trusts at the end of the first year is \(\$190,740.57\). How much money was invested in each trust? Set the following as equations in a matrix form. Identify the unknowns. Do not solve for the unknowns.
Answer: \(\left[\begin{matrix}1&1&1\\0&1&-3\\0.06&0.08&0.11\\\end{matrix}\right]\left[\begin{matrix}J\\C\\D\\\end{matrix}\right]=\left[\begin{matrix}2,253,453\\0\\190,740.57\\\end{matrix}\right]\)
(17). By any scientific method, find the second column of the inverse of
\[\begin{bmatrix} 1 & 2 & 0 \\ 4 & 5 & 0 \\ 0 & 0 & 13 \\ \end{bmatrix}\]
Answer: \(\left[\begin{matrix}0.6666\\-0.3333\\0\\\end{matrix}\right]\)
(18). Just write out the inverse of (no need to show any work) \[\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 5 \\ \end{bmatrix}\]
Answer: \(\left[\begin{matrix}1&0&0&0\\0&0.5&0&0\\0&0&0.25&0\\0&0&0&0.20\\\end{matrix}\right]\)
(19). Solve \([A][X]= [B]\) for \([X]\) if
\[[A]^{- 1} = \begin{bmatrix}
10 & - 7 & 0 \\
2 & 2 & 5 \\
2 & 0 & 6 \\
\end{bmatrix},\]
and
\[[B]= \begin{bmatrix} 7 \\ 2.5 \\ 6.012 \\ \end{bmatrix}.\]
Answer: \(\left[X\right]=\left[\begin{matrix}52.5\\49.06\\50.072\\\end{matrix}\right]\)
(20). Find the missing coefficients in the following matrices.
\[\begin{bmatrix} 1 & 2 & 0 \\ 4 & 5 & 0 \\ 0 & 0 & 13 \\ \end{bmatrix}\begin{bmatrix} X & a_{12}^{\prime} & X \\ X & a_{22}^{\prime} & X \\ X & a_{32}^{\prime} & X \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\]
Answer: \([A]^{-1} = \begin{bmatrix} -1.666 & 0.6666 & 0 \\ 1.333 & -0.3333 & 0 \\ 0 & 0 & 0.07692 \end{bmatrix}\)