Q8. - Algebra 2

Question

Use matrices A, B, and C  to find the following.

$A=\left[\begin{array}{cc}2& 3\\ 5& 6\end{array}\right],B=\left[\begin{array}{cc}-1& 7\\ 0& 4\end{array}\right],C=\left[\begin{array}{cc}9& -4\\ -6& 5\end{array}\right]$

$A+B+C$

Step-by-Step Solution

Answer

The final matrix after performing the indicated matrix operations is 

$A+B+C=\left[\begin{array}{cc}10& 6\\ -1& 15\end{array}\right]$

Step 1 - Substitute the matrices

Substitute$\left[\begin{array}{cc}2& 3\\ 5& 6\end{array}\right]$for A, $\left[\begin{array}{cc}-1& 7\\ 0& 4\end{array}\right]$for B and $\left[\begin{array}{cc}9& -4\\ -6& 5\end{array}\right]$for C into $A+B+C$

$A+B+C=\left[\begin{array}{cc}2& 3\\ 5& 6\end{array}\right]+\left[\begin{array}{cc}-1& 7\\ 0& 4\end{array}\right]+\left[\begin{array}{cc}9& -4\\ -6& 5\end{array}\right]$

Step 2 - Perform the matrix addition

Matrix addition - 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.

$\begin{array}{l}A+B+C=\left[\begin{array}{cc}2& 3\\ 5& 6\end{array}\right]+\left[\begin{array}{cc}-1& 7\\ 0& 4\end{array}\right]+\left[\begin{array}{cc}9& -4\\ -6& 5\end{array}\right]\\ =\left[\begin{array}{cc}2-1+9& 3+7-4\\ 5+0-6& 6+4+5\end{array}\right]\end{array}$

Step 3 - Simplify

Further simplify the matrix.

$\begin{array}{l}A+B+C=\left[\begin{array}{cc}2-1+9& 3+7-4\\ 5+0-6& 6+4+5\end{array}\right]\\ =\left[\begin{array}{cc}10& 6\\ -1& 15\end{array}\right]\end{array}$