Q158 / 270 · NDA 2015 (Phase 2) · Mathematics

If $A=\left[\begin{array}{ccc}1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}-1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5\end{array}\right]$ then which of the following is/are correct? 1. $A$ and $B$ commute. 2. $A B$ is a null matrix. Select the correct answer using the code given below:

  • A
    1 only
  • Correct answer
    B
    2 only
  • C
    Both 1 and 2
  • D
    Neither 1 nor 2

Solution

$\begin{aligned} & \mathrm{A}=\left[\begin{array}{ccc}1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3\end{array}\right] ; \mathrm{B}=\left[\begin{array}{ccc}-1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5\end{array}\right] \\ & \mathrm{AB}=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right] \\ & \mathrm{BA}=\left[\begin{array}{ccc}-8 & 7 & -10 \\ 48 & -42 & 60 \\ 40 & -35 & 50\end{array}\right] \\ & \text {as } \mathrm{AB} \neq \mathrm{BA} \\ & \text {So } \mathrm{A} \text { and } \mathrm{B} \text { are not commute. }\end{aligned}$

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

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