Quiz Chapter 04.01: Introduction to Matrix Algebra

1. Given $\begin{bmatrix}6&2&3&9 \\ 0&1&2&3 \\ 0&0&4&5 \\ 0&0&0&6 \end{bmatrix}$ then $\left[ A \right]$ is a(n) _____ matrix.

diagonal

identity

lower triangular

upper triangular

2. A square matrix $\left[ A \right]$ is lower triangular if

a_{ij}=0,\,j>i

a_{ij}=0,\,i>j

a_{ij} \neq 0,\,i>j

a_{ij} \neq 0,\,j>i

3. Given $\left[ A \right] = \begin{bmatrix} 12.3&-12.3&20.3 \\ 11.3&-10.3&-11.3 \\ 10.3&-11.3&-12.3 \\ \end{bmatrix}$ , $\left[ B \right] = \begin{bmatrix} 2&4 \\ -5&6 \\ 11&-20 \\ \end{bmatrix}$ then if $\left[ C \right] = \left[ A \right] \left[ B \right]$ , then $c_{31}=$ _____

-58.2

-37.6

219.4

259.4

4. The following system of equations has _____ solution(s).

- - $x + y = 2$
  - $6x+6y=12$

infinite

two

unique

5. Consider there are only two computer companies in a country. The companies are named Dude and Imac. Each year, Dude keeps $1/5$ ^th of its customers, while the rest switch to Imac. Each year, Imac keeps $1/3$ ^rd of its customers, while the rest switch to Dude. If in 2003, Dude had $1/6$ ^th of the market and Imac had $5/6$ ^th of the market, what will be the share of Dude computers when the market becomes stable?

37/90

5/11

6/11

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 and $A$ 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 $6%$ , $8%$ , and $11%$ , 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 $\left( J \right)$ , Corey $\left( C \right)$ , and David $\left( D \right)$ in a matrix form is

\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}

\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}

\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}

\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 \\ 190,740.57 \\ \end{bmatrix}