If A= [ cc 3 & -4 -1 & 2 ], find matrix B such that B A=1, where … - Vidyadip

Question

If $A=\left[\begin{array}{cc}3 & -4 \\ -1 & 2\end{array}\right]$, find matrix $B$ such that $B A=1$, where $I$ is unity matrix of order 2

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Answer

$
A=\left[\begin{array}{cc}
3 & -4 \\
-1 & 2
\end{array}\right]
$
$BA = I$, where $I$ is unity matrix of order 2
$
\therefore I=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]
$Let $B =\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$
$\therefore BA =\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \times\left[\begin{array}{cc}3 & -4 \\ -1 & 2\end{array}\right]$
$=\left[\begin{array}{ll}3 a-b & -4 a+2 b \\ 3 c-d & -4 c+2 d\end{array}\right]$
$
\therefore\left[\begin{array}{ll}
3 a-b & -4 a+2 b \\
3 c-d & -4 c+2 d
\end{array}\right]=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]
$
Comparing the corresponding terms, we get
$
\begin{aligned}
& 3 a-b=1, \\
& -4 a+2 b=0 \\
& \Rightarrow 2 b=4 a \\
& \Rightarrow b=2 a \\
& \therefore 3 a-b=1 \\
& \Rightarrow 3 a-2 a=1 \\
& \Rightarrow a=1 \\
& \text { and } \\
& b=2 a \\
& \Rightarrow b=2 \times 1=2 \\
& \therefore a=1, b=2
\end{aligned}
$
and
$3 c-d=0$
$\Rightarrow d=3 c$
$-4 c +2 d =1$
$\Rightarrow-4 c+2 \times 3 c=1$
$\Rightarrow-4 c+6 c=1$
$\Rightarrow 2 x =1$
$\Rightarrow c=\frac{1}{2}$
and
$d =3 c =3 \times \frac{1}{2}=\frac{3}{2}$
Hence $a=1, b=2, c=\frac{1}{2}, d=\frac{3}{2}$
$\therefore$ Matrix $B=\left[\begin{array}{ll}1 & 2 \\ \frac{1}{2} & \frac{3}{2}\end{array}\right]$.

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