Chapter 10.4, Problem 17E

To determine

To calculate:

The solution of given matrix equation:

  $2\left(B-X\right)=D$

Answer to Problem 17E

There is no matrix solution.

Explanation of Solution

Given information:

  $\begin{array}{l}A=\left[\begin{array}{c}4\\ 1\end{array}\begin{array}{c}6\\ 3\end{array}\right]\\ B=\left[\begin{array}{c}2\\ 3\end{array}\begin{array}{c}5\\ 7\end{array}\right]\\ C=\left[\begin{array}{l}\begin{array}{cc}2& 3\end{array}\\ \begin{array}{cc}1& 0\end{array}\\ \begin{array}{cc}0& 2\end{array}\end{array}\right]\\ D=\left[\begin{array}{l}\begin{array}{cc}10& 20\end{array}\\ \begin{array}{cc}30& 20\end{array}\\ \begin{array}{cc}10& 0\end{array}\end{array}\right]\end{array}$

Know that:

Properties of Matrices:

  $A+B=0$

Consider the following matrixes:

  $\begin{array}{l}A=\left[\begin{array}{c}4\\ 1\end{array}\begin{array}{c}6\\ 3\end{array}\right]\\ B=\left[\begin{array}{c}2\\ 3\end{array}\begin{array}{c}5\\ 7\end{array}\right]\\ C=\left[\begin{array}{l}\begin{array}{cc}2& 3\end{array}\\ \begin{array}{cc}1& 0\end{array}\\ \begin{array}{cc}0& 2\end{array}\end{array}\right]\\ D=\left[\begin{array}{l}\begin{array}{cc}10& 20\end{array}\\ \begin{array}{cc}30& 20\end{array}\\ \begin{array}{cc}10& 0\end{array}\end{array}\right]\end{array}$

Here determines the unknown matrix $X$ from the equation.

  $2\left(B-X\right)=D$

Here the dimensions of the matrix $B$ is $\left(2\times 2\right)$ then the matrix $X$ will also be a $\left(2\times 2\right)$ . The reason is two matrices can subtract only when they have the same dimensions.

Now use the properties of matrices as the original equation is,

  $2\left(B-X\right)=D$ .

After multiply by both $\frac{1}{2}$ on both sides and subtract $B$ the original equation $-X=\frac{1}{2}D-B$ .

Now the dimensions of the matrix $B$ is $2\times 2$ and the dimensions of the matrix $D$ is $3\times 2$ . Two matrices can add or subtract only if they have the same dimensions.

So, the two matrices $B$ and $C$ have different dimensions, so they cannot be added and $X$ cannot be added.