Perform the indicated matrix operations. 7.

$\frac{2}{3}\left[\begin{array}{cc}9& 0\\ 12& 15\end{array}\right]+\left[\begin{array}{cc}-2& 3\\ -7& -7\end{array}\right]$

A rectangular array of numbers arranged in rows and columns is called a matrix.

The matrix operation to be performed is addition.

To evaluate this matrix first perform scalar multiplication and then subtraction.

Scalar multiplication is the method in which matrix is multiplied by a constant called scalar.

Here for the matrix $\frac{2}{3}\left[\begin{array}{cc}9& 0\\ 12& 15\end{array}\right]$, $\frac{2}{3}$ is a scalar quantity.

Multiply each element of the this matrix by $\frac{2}{3}$.

That is equal to,

$\left[\begin{array}{cc}9\times \frac{2}{3}& 0\times \frac{2}{3}\\ 12\times \frac{2}{3}& 15\times \frac{2}{3}\end{array}\right]=\left[\begin{array}{cc}6& 0\\ 8& 10\end{array}\right]$

Now, Matrices can be added if they are of the same dimension.

Here, both the matrices are of dimension $(2\times 2)$ that is both are of same dimension. Then addition can be performed.

Add the corresponding elements

$\begin{array}{c}\frac{2}{3}\left[\begin{array}{cc}9& 0\\ 12& 15\end{array}\right]+\left[\begin{array}{cc}-2& 3\\ -7& -7\end{array}\right]=\left[\begin{array}{cc}6& 0\\ 8& 10\end{array}\right]+\left[\begin{array}{cc}-2& 3\\ -7& -7\end{array}\right]\\ =\left[\begin{array}{cc}6+(-2)& 0+3\\ 8+(-7)& 10+(-7)\end{array}\right]\\ =\left[\begin{array}{cc}6-2& 3\\ 8-7& 10-7\end{array}\right]\\ =\left[\begin{array}{cc}4& 3\\ 1& 3\end{array}\right]\end{array}$

Therefore, the resultant matrix is$\left[\begin{array}{cc}4& 3\\ 1& 3\end{array}\right]$