Question

If ${\left\{\begin{array}{c}\left[\begin{array}{ccc}5& 1& 4\\ 7& 6& 2\\ 1& 3& 5\end{array}\right]\left[\begin{array}{ccc}1& 6& -7\\ 6& 2& 4\\ -7& 4& 3\end{array}\right]\left[\begin{array}{ccc}5& 7& 1\\ 1& 6& 3\\ 4& 2& 5\end{array}\right]\end{array}\right\}}^{2020}=$ $\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]$ , 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, $X={\left(AB{A}^T\right)}^{2020}$
( $\because$ $B$ is symmetric)
$X^T={\left({\left(AB{A}^T\right)}^{2020}\right)}^T={\left({\left(AB{A}^T\right)}^T\right)}^{2020}={\left(AB{A}^T\right)}^{2020}$
$X=X^T\Rightarrow X$ is a symmetric matrix
$\Rightarrow a_2=b_1,a_3=c_1,b_3=c_2$
$\Rightarrow$ Required value $=0$