Question

Perform the indicated matrix operations. 7.

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

Short answer

The answer is$\left[\begin{array}{cc}4& 3\\ 1& 3\end{array}\right]$

Step-by-step solution

- State the definition of a matrix.

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

- Solve the 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

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

- State the conclusion.

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