Answer
The value of $ AB-4A $ is $\left[ \begin{matrix}
-2 & 10 \\
-5 & 7 \\
15 & -15 \\
\end{matrix} \right]$.
Work Step by Step
We have to calculate the product of AB as given below:
$\begin{align}
& AB=\left[ \begin{matrix}
4 & 2 \\
1 & -1 \\
0 & 5 \\
\end{matrix} \right]\left[ \begin{matrix}
2 & 4 \\
3 & 1 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
4\cdot 2+2\cdot 3 & 4\cdot 4+2\cdot 1 \\
1\cdot 2+\left( -1 \right)\cdot 3 & 1\cdot 4+\left( -1 \right)\cdot 1 \\
0\cdot 2+5\cdot 3 & 0\cdot 4+5\cdot 1 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
8+6 & 16+2 \\
2-3 & 4-1 \\
0+15 & 0+5 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
14 & 18 \\
-1 & 3 \\
15 & 5 \\
\end{matrix} \right]
\end{align}$
Then, for the value of $4A $ is:
$\begin{align}
& 4A=4\left[ \begin{matrix}
4 & 2 \\
1 & -1 \\
0 & 5 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
4\cdot 4 & 4\cdot 2 \\
4\cdot 1 & 4\cdot \left( -1 \right) \\
4\cdot 0 & 4\cdot 5 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
16 & 8 \\
4 & -4 \\
0 & 20 \\
\end{matrix} \right]
\end{align}$
Next, for the value of the given expression $ AB-4A $:
$\begin{align}
& AB-4A=\left[ \begin{matrix}
14 & 18 \\
-1 & 3 \\
15 & 5 \\
\end{matrix} \right]-\left[ \begin{matrix}
16 & 8 \\
4 & -4 \\
0 & 20 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
14-16 & 18-8 \\
-1-4 & 3+4 \\
15-0 & 5-20 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
-2 & 10 \\
-5 & 7 \\
15 & -15 \\
\end{matrix} \right]
\end{align}$
Thus, the value of $ AB-4A $ is $\left[ \begin{matrix}
-2 & 10 \\
-5 & 7 \\
15 & -15 \\
\end{matrix} \right]$.
