Chapter EP, Problem 4.2.5EP

To determine

The performance of indicated matrix operation.

Answer to Problem 4.2.5EP

  $\left[\begin{array}{cc}46& -30\\ -1& 20\end{array}\right]$ .

Explanation of Solution

Given information:

Perform the indicated matrix operations. If the matrix does not exist, write impossible.

  $5\left[\begin{array}{cc}6& -2\\ 5& 4\end{array}\right]-2\left[\begin{array}{cc}6& -2\\ 5& 4\end{array}\right]+4\left[\begin{array}{cc}7& -6\\ -4& 2\end{array}\right]$

Calculation:

The given matrix:

  $5\left[\begin{array}{cc}6& -2\\ 5& 4\end{array}\right]-2\left[\begin{array}{cc}6& -2\\ 5& 4\end{array}\right]+4\left[\begin{array}{cc}7& -6\\ -4& 2\end{array}\right]$

To add or subtract the two matrices with the same dimensions, add or subtract their corresponding elements are as shown below:

The given matrix is a multiple step operations. Start with distributing the scalar in each matrix.

  $\begin{array}{l}5\left[\begin{array}{cc}6& -2\\ 5& 4\end{array}\right]-2\left[\begin{array}{cc}6& -2\\ 5& 4\end{array}\right]+4\left[\begin{array}{cc}7& -6\\ -4& 2\end{array}\right]\text{ [Distributing the scalar]}\\ \left[\begin{array}{cc}6(5)& -2(5)\\ 5(5)& 4(5)\end{array}\right]-\left[\begin{array}{cc}6(2)& -2(2)\\ 5(2)& 4(2)\end{array}\right]+\left[\begin{array}{cc}7(4)& -6(4)\\ -4(4)& 2(4)\end{array}\right]\text{ [Multiply]}\\ \left[\begin{array}{cc}30& -10\\ 25& 20\end{array}\right]-\left[\begin{array}{cc}12& -4\\ 10& 8\end{array}\right]+\left[\begin{array}{cc}28& -24\\ -16& 8\end{array}\right]\end{array}$

Now add or subtract the two matrices with the same dimensions by their corresponding elements:

  $\left[\begin{array}{cc}30& -10\\ 25& 20\end{array}\right]-\left[\begin{array}{cc}12& -4\\ 10& 8\end{array}\right]+\left[\begin{array}{cc}28& -24\\ -16& 8\end{array}\right]$

  $\begin{array}{l}\left[\begin{array}{cc}30-12+28& -10-(-4)+(-24)\\ 25-10+(-16)& 20-8+8\end{array}\right]\\ \left[\begin{array}{cc}46& -30\\ -1& 20\end{array}\right]\end{array}$

Therefore, the solution is $\left[\begin{array}{cc}46& -30\\ -1& 20\end{array}\right]$ .