Let $A=\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)$ and $B=\left(\begin{array}{ll}a & 0 \\ 0 &…

Let $A=\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)$ and $B=\left(\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right), a, b \in N$. Then
  1. there cannot exist any B such that AB = BA
  2. there exist more than one but finite number of B’s such that AB = BA
  3. there exists exactly one B such that AB = BA
  4. there exist infinitely many B’s such that AB = BA

Solution

$A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] \quad B=\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right]$ $A B=\left[\begin{array}{ll}a & 2 b \\ 3 a & 4 b\end{array}\right]$ $B A=\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]=\left[\begin{array}{cc}a & 2 a \\ 3 b & 4 b\end{array}\right]$ $A B=B A$ only when $a=b$

Asked in: JEE Main 2006

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