$A=\left[\begin{array}{lll}0 & 1 & 2 \ 2 & 3 & 0 \ 4 & 0 & 3\end{array} ight]$ and $B$ is a matrix such that $A B=B A$. If $A B$ is not an identity matrix, then the matrix that can be taken as B is

Question

$A=\left[\begin{array}{lll}0 & 1 & 2 \ 2 & 3 & 0 \ 4 & 0 & 3\end{array}ight]$ and $B$ is a matrix such that $A B=B A$. If $A B$ is not an identity matrix, then the matrix that can be taken as B is

Aicharya · Accepted Answer

$\mathrm{A}=\left[\begin{array}{lll} 0 & 1 & 2 \\ 2 & 3 & 0 \\ 4 & 0 & 3 \end{array}\right] \text { and let } \mathrm{B}=\left[\begin{array}{ccc} x & y & z \\ a & b & c \\ u & v & w \end{array}\right]$
Now $\mathrm{AB}=\mathrm{BA}$ $\begin{aligned} & \Rightarrow\left[\begin{array}{lll} 0 & 1 & 2 \\ 2 & 3 & 0 \\ 4 & 0 & 3 \end{array}\right]\left[\begin{array}{lll} x & y & z \\ a & b & c \\ u & v & w \end{array}\right]=\left[\begin{array}{lll} x & y & z \\ a & b & c \\ u & v & w \end{array}\right]\left[\begin{array}{lll} 0 & 1 & 2 \\ 2 & 3 & 0 \\ 4 & 0 & 3 \end{array}\right] \\ & \Rightarrow\left[\begin{array}{ccc} a+2 u & b+2 v & c+2 w \\ 2 x+3 a & 2 y+3 b & 2 z+3 c \\ 4 x+4 v & 4 y+3 v & 4 z+3 w \end{array}\right]=\left[\begin{array}{ccc} 2 y+4 z & x+3 y & 2 x+3 z \\ 2 b+4 c & a+3 b & 2 a+3 c \\ 2 v+4 w & u+3 v & 2 u+3 w \end{array}\right] \end{aligned}$
From the given options option (4) $\left[\begin{array}{ccc}9 & -3 & -6 \\ -6 & -8 & 4 \\ -12 & 4 & -2\end{array}\right]$ satisfies the above condition.

$A=\left[\begin{array}{lll}0 & 1 & 2 \\ 2 & 3 & 0 \\ 4 & 0 & 3\end{array}\right]$ and $B$ is a matrix such…

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