Q43 / 160 · AP EAMCET 2022 (06 Jul Shift 1) · Mathematics

If $\mathrm{A}=\left[\begin{array}{ll}3 & 4 \\ 5 & 6\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}x & 0 \\ 0 & y\end{array}\right], \mathrm{x}, \mathrm{y} \in \mathrm{N}$, then

  • A
    There is exactly one such matrix $B$ such that $A B=I$
  • B
    There is no matrix $\mathrm{B}$ such that $\mathrm{AB}=\mathrm{BA}$
  • C
    There exist only a finite number of matrices $B$ such that $\mathrm{AB}=\mathrm{BA}$
  • Correct answer
    D
    There exist infinite number of matrices $B$ such that $\mathrm{AB}=\mathrm{BA}$

Solution

We have $$ \begin{aligned} & \mathrm{A}=\left[\begin{array}{ll} 3 & 4 \\ 5 & 6 \end{array}\right] \mathrm{B}=\left[\begin{array}{ll} \mathrm{x} & 0 \\ 0 & \mathrm{y} \end{array}\right] \mathrm{x}, \mathrm{y} \varepsilon \mathrm{n} \\ & \Rightarrow \mathrm{AB}=\left[\begin{array}{ll} 3 & 4 \\ 5 & 6 \end{array}\right]\left[\begin{array}{ll} \mathrm{x} & 0 \\ 0 & \mathrm{y} \end{array}\right] \\ & =\mathrm{AB}=\left[\begin{array}{ll} 3 \mathrm{x} & 4 \mathrm{y} \\ 5 \mathrm{x} & 6 \mathrm{y} \end{array}\right] \\ & =\mathrm{xy}\left[\begin{array}{ll} 3 \mathrm{x} & 4 \mathrm{y} \\ 5 \mathrm{x} & 6 \mathrm{y} \end{array}\right] \\ & \text { now } \mathrm{BA}=\mathrm{xy}\left[\begin{array}{ll} 3 & 4 \\ 5 & 6 \end{array}\right] \\ & \text { Thus } \mathrm{AB}=\mathrm{BA} \end{aligned} $$

Hence, the correct answer is option: 'D'.

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