Chapter 11.4, Problem 23AYU

To determine

To calculate: The value of the expression $CA-CB$ .

Answer to Problem 23AYU

The value of the expression $CA-CB$ is $\left[\begin{array}{ccc}-13& 7& -12\\ -18& 10& -14\\ 17& -7& 34\end{array}\right]$ .

Explanation of Solution

Given information:

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

Formula used:

To find the product AB of two matrices A and B , the number of column in matrix A must be equal to number of rows in matrixB .

Calculation:

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

First compute, $CA$ . Recall that to find the product AB of two matrices A and B , the number of column in matrix A must be equal to number of rows in matrixB .

  $\begin{array}{c}CA=\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right]\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right]\\ =\left[\begin{array}{ccc}0+1& 12+2& -20+6\\ 0+2& 18+4& -30+12\\ 3& -6+6& 10+18\end{array}\right]\\ =\left[\begin{array}{ccc}1& 14& -14\\ 2& 22& -18\\ 3& 0& 28\end{array}\right]\end{array}$

Now, compute $CB$ .

  $\begin{array}{c}CB=\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right]\left[\begin{array}{ccc}4& 1& 0\\ -2& 3& -2\end{array}\right]\\ =\left[\begin{array}{ccc}16-2& 4+3& 0-2\\ 24-4& 6+6& 0-4\\ -8-6& -2+9& 0-6\end{array}\right]\\ =\left[\begin{array}{ccc}14& 7& -2\\ 20& 12& -4\\ -14& 7& -6\end{array}\right]\end{array}$

Next, compute $CA-CB$ ,

  $\begin{array}{c}CA-CB=\left[\begin{array}{ccc}1& 14& -14\\ 2& 22& -18\\ 3& 0& 28\end{array}\right]-\left[\begin{array}{ccc}14& 7& -2\\ 20& 12& -4\\ -14& 7& -6\end{array}\right]\\ =\left[\begin{array}{ccc}-13& 7& -12\\ -18& 10& -14\\ 17& -7& 34\end{array}\right]\end{array}$

Thus, the value of the expression $CA-CB$ is $\left[\begin{array}{ccc}-13& 7& -12\\ -18& 10& -14\\ 17& -7& 34\end{array}\right]$ .