Q12 / 75 · NTA Abhyas (JEE Main) 2020 · Mathematics · Matrices

Let $A = [\begin{matrix} 2 & 3 \\ 5 & 7 \end{matrix}]$ and $B = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}]$ where $a, b \in N$ . The number of matrices $B$ such that $A B = B A,$ is equal to

A
$0$
B
$1$
C
$2$
Correct answer
D
infinite

Solution

A B = [\begin{matrix} 2 & 3 \\ 5 & 7 \end{matrix}] [\begin{matrix} a & 0 \\ 0 & b \end{matrix}] = [\begin{matrix} 2 a & 3 b \\ 5 a & 7 b \end{matrix}]

B A = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}] [\begin{matrix} 2 & 3 \\ 5 & 7 \end{matrix}] = [\begin{matrix} 2 a & 3 a \\ 5 b & 7 b \end{matrix}]

∴ A B = B A \Rightarrow a = b

(infinite matrices are possible)

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

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