Question

Perform the indicated matrix operation. If the matrix doesn’t exist, write impossible.

19.$5\left[\begin{array}{c}1\\ -1\\ -3\end{array}\right]+6\left[\begin{array}{c}-4\\ 3\\ 5\end{array}\right]-2\left[\begin{array}{c}-3\\ 8\\ -4\end{array}\right]$

Short answer

The final matrix after performing the indicated matrix operations is

$5\left[\begin{array}{c}1\\ -1\\ -3\end{array}\right]+6\left[\begin{array}{c}-4\\ 3\\ 5\end{array}\right]-2\left[\begin{array}{c}-3\\ 8\\ -4\end{array}\right]=\left[\begin{array}{c}-13\\ -3\\ 23\end{array}\right]$.

Step-by-step solution

- Define the concept used

Matrix addition/subtraction and scalar multiplication:

If A and B are two matrices of order $m\times n$, then the addition/subtraction of the two matrices will also yield a matrix of order $m\times n$wherein each element is the sum/difference of the corresponding elements.

The product of a scalar k and an $m\times n$ matrix is an $m\times n$ matrix in which each element equals k times the corresponding elements of the original matrix.

- Perform the scalar multiplication first

Multiply each element in the first matrix $\left[\begin{array}{c}1\\ -1\\ -3\end{array}\right]$by 5, multiply each element in the second matrix $\left[\begin{array}{c}-4\\ 3\\ 5\end{array}\right]$by 6 and multiply each element in the third matrix $\left[\begin{array}{c}-3\\ 8\\ -4\end{array}\right]$by -2.

$\begin{array}{c}5\left[\begin{array}{c}1\\ -1\\ -3\end{array}\right]+6\left[\begin{array}{c}-4\\ 3\\ 5\end{array}\right]-2\left[\begin{array}{c}-3\\ 8\\ -4\end{array}\right]=\left[\begin{array}{c}5\left(1\right)\\ 5\left(-1\right)\\ 5\left(-3\right)\end{array}\right]+\left[\begin{array}{c}6\left(-4\right)\\ 6\left(3\right)\\ 6\left(5\right)\end{array}\right]+\left[\begin{array}{c}-2\left(-3\right)\\ -2\left(8\right)\\ -2\left(-4\right)\end{array}\right]\\ =\left[\begin{array}{c}5\\ -5\\ -15\end{array}\right]+\left[\begin{array}{c}-24\\ 18\\ 30\end{array}\right]+\left[\begin{array}{c}6\\ -16\\ 8\end{array}\right]\end{array}$

- Add corresponding elements

Add corresponding elements of both the matrices $\left[\begin{array}{c}5\\ -5\\ -15\end{array}\right]+\left[\begin{array}{c}-24\\ 18\\ 30\end{array}\right]+\left[\begin{array}{c}6\\ -16\\ 8\end{array}\right]$and simplify.

$\begin{array}{c}5\left[\begin{array}{c}1\\ -1\\ -3\end{array}\right]+6\left[\begin{array}{c}-4\\ 3\\ 5\end{array}\right]-2\left[\begin{array}{c}-3\\ 8\\ -4\end{array}\right]=\left[\begin{array}{c}5\\ -5\\ -15\end{array}\right]+\left[\begin{array}{c}-24\\ 18\\ 30\end{array}\right]+\left[\begin{array}{c}6\\ -16\\ 8\end{array}\right]\\ =\left[\begin{array}{c}5-24+6\\ -5+18-16\\ -15+30+8\end{array}\right]\\ =\left[\begin{array}{c}-13\\ -3\\ 23\end{array}\right]\end{array}$