Q54. - Algebra 1

Question

Perform the indicated matrix operation.

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

Step-by-Step Solution

Answer

The addition of the matrices is $\mathbf{3}\left[\begin{array}{cccc}5& -3& -1& 4\\ 7& -6& 8& 1\end{array}\right]\mathbf{=}\left[\begin{array}{cccc}15& -9& -3& 12\\ 21& -18& 24& 3\end{array}\right]$.

Step 1. Define the concept of scalar multiplication.

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.

Step 2. Multiply all the elements of the matrix with scalar 3.

In order to find $3\left[\begin{array}{cccc}5& -3& -1& 4\\ 7& -6& 8& 1\end{array}\right]$, multiply each element of the matrix by 3.

 $3\left[\begin{array}{cccc}5& -3& -1& 4\\ 7& -6& 8& 1\end{array}\right]=\left[\begin{array}{cccc}3\left(5\right)& 3(-3)& 3(-1)& 3\left(4\right)\\ 3\left(7\right)& 3(-6)& 3\left(8\right)& 3\left(1\right)\end{array}\right]$

Step 3. Simplify the matrix.

Further simplify the matrix by multiplying the elements.

$\begin{array}{c}3\left[\begin{array}{cccc}5& -3& -1& 4\\ 7& -6& 8& 1\end{array}\right]=\left[\begin{array}{cccc}3\left(5\right)& 3(-3)& 3(-1)& 3\left(4\right)\\ 3\left(7\right)& 3(-6)& 3\left(8\right)& 3\left(1\right)\end{array}\right]\\ =\left[\begin{array}{cccc}15& -9& -3& 12\\ 21& -18& 24& 3\end{array}\right]\end{array}$

Therefore, $3\left[\begin{array}{cccc}5& -3& -1& 4\\ 7& -6& 8& 1\end{array}\right]=\left[\begin{array}{cccc}15& -9& -3& 12\\ 21& -18& 24& 3\end{array}\right]$.