Question

If $\begin{Bmatrix} \begin{bmatrix} 5 & 1 & 4 \\ 7 & 6 & 2 \\ 1 & 3 & 5 \end{bmatrix} \begin{bmatrix} 1 & 6 & -7 \\ 6 & 2 & 4 \\ -7 & 4 & 3 \end{bmatrix} \begin{bmatrix} 5 & 7 & 1 \\ 1 & 6 & 3 \\ 4 & 2 & 5 \end{bmatrix} \end{Bmatrix}^{2020}=$ $\begin{bmatrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{bmatrix}$ , then the value of $2\left|a_{2} - b_{1}\right|+3\left|a_{3} - c_{1}\right|+4\left|b_{3} - c_{2}\right|$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (A)

Let, $X=\left(A B A^{T}\right)^{2020}$
( $\because $ $B$ is symmetric)
$X^{T}=\left(\left(A B A^{T}\right)^{2020}\right)^{T}=\left(\left(A B A^{T}\right)^{T}\right)^{2020}=\left(A B A^{T}\right)^{2020}$
$X=X^{T}\Rightarrow X$ is a symmetric matrix
$\Rightarrow a_{2}=b_{1},a_{3}=c_{1},b_{3}=c_{2}$
$\Rightarrow $ Required value $=0$