Answer
a) $A+B=\left[ \begin{matrix}
9 & 10 \\
3 & 9 \\
\end{matrix} \right]$
b) $A-B=\left[ \begin{matrix}
-1 & -8 \\
3 & -5 \\
\end{matrix} \right]$
c) $\left( -4 \right)A=\left[ \begin{matrix}
-16 & -4 \\
-12 & -8 \\
\end{matrix} \right]$
d) $3A+2B=\left[ \begin{matrix}
22 & 21 \\
9 & 20 \\
\end{matrix} \right]$
Work Step by Step
(a)
Perform the addition of the matrices $A$ and $B$ as follows:
$\begin{align}
& A+B=\left[ \begin{matrix}
4 & 1 \\
3 & 2 \\
\end{matrix} \right]+\left[ \begin{matrix}
5 & 9 \\
0 & 7 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
4+5 & 1+9 \\
3+0 & 2+7 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
9 & 10 \\
3 & 9 \\
\end{matrix} \right]
\end{align}$
Hence, $A+B=\left[ \begin{matrix}
9 & 10 \\
3 & 9 \\
\end{matrix} \right]$
(b)
Perform the subtraction of the matrices $A$ and $B$ as follows:
$\begin{align}
& A-B=\left[ \begin{matrix}
4 & 1 \\
3 & 2 \\
\end{matrix} \right]-\left[ \begin{matrix}
5 & 9 \\
0 & 7 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
4-5 & 1-9 \\
3-0 & 2-7 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
-1 & -8 \\
3 & -5 \\
\end{matrix} \right]
\end{align}$
Hence, $A-B=\left[ \begin{matrix}
-1 & -8 \\
3 & -5 \\
\end{matrix} \right]$
(c)
Perform the multiplication of the matrices $A$ with constant number.
$\begin{align}
& \left( -4 \right)A=\left( -4 \right)\left[ \begin{matrix}
4 & 1 \\
3 & 2 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
-16 & -4 \\
-12 & -8 \\
\end{matrix} \right]
\end{align}$
Hence, $\left( -4 \right)A=\left[ \begin{matrix}
-16 & -4 \\
-12 & -8 \\
\end{matrix} \right]$
(d)
Perform the multiplication with a content number and addition of the matrices $A$ and $B$ as follows:
$\begin{align}
& 3A+2B=3\left[ \begin{matrix}
4 & 1 \\
3 & 2 \\
\end{matrix} \right]+2\left[ \begin{matrix}
5 & 9 \\
0 & 7 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
3\times 4 & 3\times 1 \\
3\times 3 & 3\times 2 \\
\end{matrix} \right]+\left[ \begin{matrix}
2\times 5 & 2\times 9 \\
2\times 0 & 2\times 7 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
12 & 3 \\
9 & 6 \\
\end{matrix} \right]+\left[ \begin{matrix}
10 & 18 \\
0 & 14 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
22 & 21 \\
9 & 20 \\
\end{matrix} \right]
\end{align}$
Hence, $3A+2B=\left[ \begin{matrix}
22 & 21 \\
9 & 20 \\
\end{matrix} \right]$
