Q18. - Algebra 2

Question

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

 

18. $5\left[\begin{array}{cccc}0& -1& 7& 2\end{array}\right]+3\left[\begin{array}{cccc}5& -8& 10& -4\end{array}\right]$

Step-by-Step Solution

Answer

The final matrix after performing the indicated matrix operations is

$5\left[\begin{array}{cccc}0& -1& 7& 2\end{array}\right]+3\left[\begin{array}{cccc}5& -8& 10& -4\end{array}\right]=\left[\begin{array}{cccc}15& -29& 65& -2\end{array}\right]$.

Step 1 - Define the concept used

Matrix addition and scalar multiplication:

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

Step 2 - Perform the scalar multiplication first

Multiply each element in the first matrix $\left[\begin{array}{cccc}0& -1& 7& 2\end{array}\right]$ by 5 and multiply each element in the second matrix $\left[\begin{array}{cccc}5& -8& 10& -4\end{array}\right]$ by 3.

$\begin{array}{c}5\left[\begin{array}{cccc}0& -1& 7& 2\end{array}\right]+3\left[\begin{array}{cccc}5& -8& 10& -4\end{array}\right]=\left[\begin{array}{cccc}5\left(0\right)& 5\left(-1\right)& 5\left(7\right)& 5\left(2\right)\end{array}\right]+\left[\begin{array}{cccc}3\left(5\right)& 3\left(-8\right)& 3\left(10\right)& 3\left(-4\right)\end{array}\right]\\ =\left[\begin{array}{cccc}0& -5& 35& 10\end{array}\right]+\left[\begin{array}{cccc}15& -24& 30& -12\end{array}\right]\end{array}$

Step 3 - Add corresponding elements

Add corresponding elements of both the matrices $\left[\begin{array}{cccc}0& -5& 35& 10\end{array}\right]+\left[\begin{array}{cccc}15& -24& 30& -12\end{array}\right]$ and simplify.

$\begin{array}{c}5\left[\begin{array}{cccc}0& -1& 7& 2\end{array}\right]+3\left[\begin{array}{cccc}5& -8& 10& -4\end{array}\right]=\left[\begin{array}{cccc}0& -5& 35& 10\end{array}\right]+\left[\begin{array}{cccc}15& -24& 30& -12\end{array}\right]\\ =\left[\begin{array}{cccc}0+15& -5-24& 35+30& 10-12\end{array}\right]\\ =\left[\begin{array}{cccc}15& -29& 65& -2\end{array}\right]\end{array}$