Q16. - Algebra 1

Question

Perform the indicated matrix operations. If an operation cannot be performed, write impossible.

$\left[\begin{array}{ccc}1& 8& -4\\ -2& 6& 7\\ 0& 9& -6\end{array}\right]\mathbf{-}\left[\begin{array}{ccc}-4& 2& 0\\ -5& 3& 8\\ -9& -7& 1\end{array}\right]$

Step-by-Step Solution

Answer

The result after performing the indicated matrix operations is $\left[\begin{array}{ccc}5& 6& -4\\ 3& 3& -1\\ 9& 16& -7\end{array}\right]$.

Step 1. Write the condition for the subtraction of two matrices.

The two matrices $A$ and $B$ can be subtracted if both matrices $A$ and $B$ have same order.

The matrix obtained after the subtraction of the two matrices has the same order as that of the matrices which are subtracted and the matrix after the subtraction can be obtained by subtracting the corresponding elements of the two matrices.

Step 2. Add the given matrices.

The order of the matrix $\left[\begin{array}{ccc}1& 8& -4\\ -2& 6& 7\\ 0& 9& -6\end{array}\right]$ is  $3\times 3$.

The order of the matrix $\left[\begin{array}{ccc}-4& 2& 0\\ -5& 3& 8\\ -9& -7& 1\end{array}\right]$ is $3\times 3$.

As, the order of the matrices is same. Therefore, the given matrices can be subtracted and the subtraction of two matrices can be obtained by subtracting the corresponding elements of the given matrices.

The result after the subtraction of the given matrices is:

$\begin{array}{c}\left[\begin{array}{ccc}1& 8& -4\\ -2& 6& 7\\ 0& 9& -6\end{array}\right]-\left[\begin{array}{ccc}-4& 2& 0\\ -5& 3& 8\\ -9& -7& 1\end{array}\right]=\left[\begin{array}{ccc}1-(-4)& 8-2& -4-0\\ -2-(-5)& 6-3& 7-8\\ 0-(-9)& 9-(-7)& -6-1\end{array}\right]\\ =\left[\begin{array}{ccc}1+4& 6& -4\\ -2+5& 3& -1\\ 0+9& 9+7& -7\end{array}\right]\\ =\left[\begin{array}{ccc}5& 6& -4\\ 3& 3& -1\\ 9& 16& -7\end{array}\right]\end{array}$

Therefore, the result after performing the indicated matrix operations is $\left[\begin{array}{ccc}5& 6& -4\\ 3& 3& -1\\ 9& 16& -7\end{array}\right]$.