Q44 / 160 · TS EAMCET 2022 (19 Jul Shift 1) · Mathematics
If $\left[\begin{array}{ccc}-1 & 2 & b \\ a & 5 & 6 \\ 3 & c & 7\end{array}\right]$ is a symmetric matrix, then $\left[\begin{array}{ccc}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]=$
- A0
- B-121
- C143
- Correct answerD-143
Solution
Given matrix is $\left[\begin{array}{ccc}-1 & 2 & b \\ a & 5 & 6 \\ 3 & c & 7\end{array}\right]$ and it is symmetric matrix.
Let $X=\left[\begin{array}{ccc}-1 & 2 & b \\ a & 5 & 6 \\ 3 & c & 7\end{array}\right]$
So, $X=X^{\prime}$
$\left[\begin{array}{ccc}
-1 & 2 & b \\
a & 5 & 6 \\
3 & c & 7
\end{array}\right]=\left[\begin{array}{ccc}
-1 & a & 3 \\
2 & 5 & c \\
b & 6 & 7
\end{array}\right]$
Compare the above matrices.
$a=2, b=3, c=6$
Then, $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=\left|\begin{array}{lll}2 & 3 & 6 \\ 3 & 6 & 2 \\ 6 & 2 & 3\end{array}\right|$
Expand w.r.t. $\mathrm{R}_1$.
$\begin{aligned}
& \Rightarrow 2(18-4)-3(9-12)+6(6-36) \\
& \Rightarrow 2 \times 14-3 \times(-3)+6 \times(-30) \\
& \Rightarrow 28+9-180 \\
& \Rightarrow 37-180=-143
\end{aligned}$
Therefore, the value of determinant is -143 .
Hence, the correct answer is option: 'D'.