Q43 / 160 · TS EAMCET 2022 (19 Jul Shift 1) · Mathematics
If $\left[\begin{array}{ccc}0 & 2 & a \\ b & 0 & 4 \\ -3 & c & 0\end{array}\right]$ is a skew-symmetric matrix, then
$\left[\begin{array}{ll}
a & b \\
b & a
\end{array}\right]\left[\begin{array}{ll}
b & c \\
c & b
\end{array}\right]=$
- A$\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$
- B$\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
- Correct answerC$\left[\begin{array}{cc}2 & -8 \\ -8 & 2\end{array}\right]$
- D$\left[\begin{array}{ll}2 & 8 \\ 8 & 2\end{array}\right]$
Solution
Let $\mathrm{A}=\left[\begin{array}{ccc}0 & 2 & a \\ b & 0 & 4 \\ -3 & c & 0\end{array}\right]$ which is skew symmetric matrix
So, $\mathrm{A}=-\mathrm{A}^{\prime}$
$\begin{aligned}
& {\left[\begin{array}{ccc}
0 & 2 & a \\
b & 0 & 4 \\
-3 & c & 0
\end{array}\right]=-\left[\begin{array}{ccc}
0 & b & -3 \\
2 & 0 & c \\
a & 4 & 0
\end{array}\right]} \\
& {\left[\begin{array}{ccc}
0 & 2 & a \\
b & 0 & 4 \\
-3 & c & 0
\end{array}\right]=\left[\begin{array}{ccc}
0 & -b & 3 \\
-2 & 0 & -c \\
-a & -4 & 0
\end{array}\right]}
\end{aligned}$
Compare the above matrices.
Then, $-b=2 \Rightarrow b=-2, a=3$ and $c=-4$.
Now,
$\begin{aligned}
& {\left[\begin{array}{ll}
a & b \\
b & a
\end{array}\right]\left[\begin{array}{ll}
b & c \\
c & b
\end{array}\right]=\left[\begin{array}{cc}
3 & -2 \\
-2 & 3
\end{array}\right]\left[\begin{array}{cc}
-2 & -4 \\
-4 & -2
\end{array}\right]} \\
& =\left[\begin{array}{cc}
-6+8 & -12+4 \\
4-12 & 8-6
\end{array}\right]=\left[\begin{array}{cc}
2 & -8 \\
-8 & 2
\end{array}\right]
\end{aligned}$
Required matrix is $\left[\begin{array}{cc}2 & -8 \\ -8 & 2\end{array}\right]$.
Hence, the correct answer is option: 'C'.