Question
If $A=\left[\begin{array}{cc}1 & 4 \\ 0 & -1\end{array}\right], B=\left[\begin{array}{cc}2 & x \\ 0 & -\frac{1}{2}\end{array}\right]$ find the value of $x$ if $A B=B A$
✓
Answer
Given
$
\begin{aligned}
& A =\left[\begin{array}{cc}
1 & 4 \\
0 & -1
\end{array}\right]\left[\begin{array}{cc}
2 & x \\
0 & -\frac{1}{2}
\end{array}\right] \\
& =\left[\begin{array}{cc}
2+0 & x-2 \\
0+0 & 0+\frac{1}{2}
\end{array}\right] \\
& =\left[\begin{array}{cc}
2 & x-2 \\
0 & \frac{1}{2}
\end{array}\right] \\
& BA =\left[\begin{array}{cc}
2^2 & x \\
0 & -\frac{1}{2}
\end{array}\right]\left[\begin{array}{cc}
1 & 4 \\
0 & -1
\end{array}\right] \\
& =\left[\begin{array}{cc}
2+0 & 8-x \\
0+0 & 0+\frac{1}{2}
\end{array}\right] \\
& =\left[\begin{array}{cc}
2 & 8-x \\
0 & \frac{1}{2}
\end{array}\right] \\
& \because AB = BA \\
& \therefore\left[\begin{array}{ll}
2 & x-2 \\
0 & \frac{1}{2}
\end{array}\right]=\left[\begin{array}{cc}
2 & 8-x \\
0 & \frac{1}{2}
\end{array}\right]
\end{aligned}
$
Comparing the corresponding elements
$
\begin{aligned}
& x-2=8-x \\
& \Rightarrow x+x=8+2 \\
& \Rightarrow 2 x=10 \\
& \therefore x=\frac{10}{2}=5 .
\end{aligned}
$
$
\begin{aligned}
& A =\left[\begin{array}{cc}
1 & 4 \\
0 & -1
\end{array}\right]\left[\begin{array}{cc}
2 & x \\
0 & -\frac{1}{2}
\end{array}\right] \\
& =\left[\begin{array}{cc}
2+0 & x-2 \\
0+0 & 0+\frac{1}{2}
\end{array}\right] \\
& =\left[\begin{array}{cc}
2 & x-2 \\
0 & \frac{1}{2}
\end{array}\right] \\
& BA =\left[\begin{array}{cc}
2^2 & x \\
0 & -\frac{1}{2}
\end{array}\right]\left[\begin{array}{cc}
1 & 4 \\
0 & -1
\end{array}\right] \\
& =\left[\begin{array}{cc}
2+0 & 8-x \\
0+0 & 0+\frac{1}{2}
\end{array}\right] \\
& =\left[\begin{array}{cc}
2 & 8-x \\
0 & \frac{1}{2}
\end{array}\right] \\
& \because AB = BA \\
& \therefore\left[\begin{array}{ll}
2 & x-2 \\
0 & \frac{1}{2}
\end{array}\right]=\left[\begin{array}{cc}
2 & 8-x \\
0 & \frac{1}{2}
\end{array}\right]
\end{aligned}
$
Comparing the corresponding elements
$
\begin{aligned}
& x-2=8-x \\
& \Rightarrow x+x=8+2 \\
& \Rightarrow 2 x=10 \\
& \therefore x=\frac{10}{2}=5 .
\end{aligned}
$
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