Chapter 11, Problem 9CT

To determine

The value of the expression is $CB$ where $A=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& 2\end{array}\right],B=\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right],C=\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right]$.

Answer to Problem 9CT

Solution:

The value of the expression $CB$ is $\left[\begin{array}{ccc}4& 10& 26\\ 1& -11& 2\\ -1& 26& 3\end{array}\right]$.

Explanation of Solution

Given information:

The matrices, $A=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& 2\end{array}\right],B=\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right],C=\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right]$

Explanation:

Consider an expression $CB$,

Here is the product of a $3\times 2$ matrix and a $2\times 3$ matrix.

The number of columns in the first matrix is the same as the number of rows in the second matrix, the operation can be performed.

$\Rightarrow CB=\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right]\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right]$

$\Rightarrow CB=\left[\begin{array}{ccc}4\cdot 1+6\cdot 0& 4\cdot \left(-2\right)+6\cdot 3& 4\cdot 5+6\cdot 1\\ 1\cdot 1+\left(-3\right)\cdot 0& 1\cdot \left(-2\right)+\left(-3\right)\cdot 3& 1\cdot 5+\left(-3\right)\cdot 1\\ \left(-1\right)\cdot 1+8\cdot 0& \left(-1\right)\cdot \left(-2\right)+8\cdot 3& \left(-1\right)\cdot 5+8\cdot 1\end{array}\right]$

$\Rightarrow CB=\left[\begin{array}{ccc}4+0& -8+18& 20+6\\ 1+0& -2-9& 5-3\\ -1+0& 2+24& -5+8\end{array}\right]$

$\Rightarrow CB=\left[\begin{array}{ccc}4& 10& 26\\ 1& -11& 2\\ -1& 26& 3\end{array}\right]$

Therefore, the value of the expression $CB$ is $\left[\begin{array}{ccc}4& 10& 26\\ 1& -11& 2\\ -1& 26& 3\end{array}\right]$.