Chapter 7.5, Problem 41E

a.

To determine

To find $AB$ and check result by matrix calculator.

Answer to Problem 41E

  $AB=\left[\begin{array}{l}015\\ 612\end{array}\right]$

Explanation of Solution

Given:

  $A=\left[\begin{array}{l}12\\ 42\end{array}\right],B=\left[\begin{array}{l}2-1\\ -18\end{array}\right]$

Concept Used:

If $A=\left[a_{ij}\right]$ is an $m\times n$ matrix and $B=\left[b_{ij}\right]$ is an $n\times p$ matrix, then the product AB is an $m\times p$ matrix given by $AB=\left[c_{ij}\right]$ , where $c_{ij}=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}\mathrm{...}+a_{in}{b}_{nj}$ .

Calculation:

Calculate the product $AB$ as shown:

  $\begin{array}{l}AB=\left[\begin{array}{l}12\\ 42\end{array}\right]\left[\begin{array}{l}2-1\\ -18\end{array}\right]\\ =\left[\begin{array}{l}1\left(2\right)+2\left(-1\right)1\left(-1\right)+2\left(8\right)\\ 4\left(2\right)+2\left(-1\right)4\left(-1\right)+2\left(8\right)\end{array}\right]\\ =\left[\begin{array}{l}2-2-1+16\\ 8-2-4+16\end{array}\right]\\ =\left[\begin{array}{l}015\\ 612\end{array}\right]\end{array}$

Now, by graphing utility of matrix capability,

  

Hence, the result is verified.

b.

To determine

To find $BA$ and check result by matrix calculator.

Answer to Problem 41E

  $BA=\left[\begin{array}{l}-22\\ 3114\end{array}\right]$

Explanation of Solution

Given:

  $A=\left[\begin{array}{l}12\\ 42\end{array}\right],B=\left[\begin{array}{l}2-1\\ -18\end{array}\right]$

Concept Used:

If $A=\left[a_{ij}\right]$ is an $m\times n$ matrix and $B=\left[b_{ij}\right]$ is an $n\times p$ matrix, then the product AB is an $m\times p$ matrix given by $AB=\left[c_{ij}\right]$ , where $c_{ij}=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}\mathrm{...}+a_{in}{b}_{nj}$ .

Calculation:

Calculate the product $BA$ as shown:

  $\begin{array}{l}BA=\left[\begin{array}{l}2-1\\ -18\end{array}\right]\left[\begin{array}{l}12\\ 42\end{array}\right]\\ =\left[\begin{array}{l}2\left(1\right)+\left(-1\right)\left(4\right)2\left(2\right)+\left(-1\right)\left(2\right)\\ -1\left(1\right)+8\left(4\right)-1\left(2\right)+8\left(2\right)\end{array}\right]\\ =\left[\begin{array}{l}2-44-2\\ -1+32-2+16\end{array}\right]\\ =\left[\begin{array}{l}-22\\ 3114\end{array}\right]\end{array}$

Now, by graphing utility of matrix capability,

  

Hence, the result is verified.

c.

To determine

To find $A^2$ and check result by matrix calculator.

Answer to Problem 41E

  $A^2=\left[\begin{array}{l}96\\ 1212\end{array}\right]$

Explanation of Solution

Given:

  $A=\left[\begin{array}{l}12\\ 42\end{array}\right],B=\left[\begin{array}{l}2-1\\ -18\end{array}\right]$

Concept Used:

If $A=\left[a_{ij}\right]$ is an $m\times n$ matrix and $B=\left[b_{ij}\right]$ is an $n\times p$ matrix, then the product AB is an $m\times p$ matrix given by $AB=\left[c_{ij}\right]$ , where $c_{ij}=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}\mathrm{...}+a_{in}{b}_{nj}$ .

Calculation:

Calculate the product $A^2$ as shown:

  $\begin{array}{l}{A}^2=AA=\left[\begin{array}{l}12\\ 42\end{array}\right]\left[\begin{array}{l}12\\ 42\end{array}\right]\\ =\left[\begin{array}{l}1\left(1\right)+2\left(4\right)1\left(2\right)+2\left(2\right)\\ 4\left(1\right)+2\left(4\right)4\left(2\right)+2\left(2\right)\end{array}\right]\\ =\left[\begin{array}{l}1+82+4\\ 4+88+4\end{array}\right]\\ =\left[\begin{array}{l}96\\ 1212\end{array}\right]\end{array}$

Now, by graphing utility of matrix capability,

  

Hence, the result is verified.