Chapter EP, Problem 4.2.13EP

To determine

The solution of the matrix operation.

Answer to Problem 4.2.13EP

  $\left[\begin{array}{cc}0& 2\\ 3& -1\end{array}\right]$ .

Explanation of Solution

Given information:

Use matrices $A,B,C,D,$ and $E$ to find the following.

  $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right],C=\left[\begin{array}{cc}2& -2\\ -3& 3\end{array}\right],D=\left[\begin{array}{cc}-2& 2\\ 3& -3\end{array}\right],E=\left[\begin{array}{cc}5& -3\\ -2& 4\end{array}\right]$

Find $D-2B$ .

Calculation:

The given matrix:

  $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right],C=\left[\begin{array}{cc}2& -2\\ -3& 3\end{array}\right],D=\left[\begin{array}{cc}-2& 2\\ 3& -3\end{array}\right],E=\left[\begin{array}{cc}5& -3\\ -2& 4\end{array}\right]$

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

Find the matrix operation $D-2B$ :

  $\begin{array}{l}D=\left[\begin{array}{cc}-2& 2\\ 3& -3\end{array}\right],B=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\text{ [The given matrix]}\\ D-2B\\ =\left[\begin{array}{cc}-2& 2\\ 3& -3\end{array}\right]-2\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\text{ [Multiplication of the matrix by 2]}\\ =\left[\begin{array}{cc}-2& 2\\ 3& -3\end{array}\right]-\left[\begin{array}{cc}-2& 0\\ 0& -2\end{array}\right]\end{array}$

  $\begin{array}{l}=\left[\begin{array}{cc}-2-(-2)& 2-0\\ 3-0& -3-(-2)\end{array}\right]\text{ [Addition of the two matrix]}\\ =\left[\begin{array}{cc}0& 2\\ 3& -1\end{array}\right]\end{array}$

Thus, the solution is $\left[\begin{array}{cc}0& 2\\ 3& -1\end{array}\right]$ .