Mathematics - Matrices Question with Solution | TestHub

Let $A$ and $B$ be real matrices such that $A=\left[\begin{array}{ll}\alpha & 0\\ 0& \beta \end{array}\right]$ and $B=\left[\begin{array}{ll}0& \gamma \\ \delta & 0\end{array}\right]$ Now, $AB=\left[\begin{array}{cc}0& \alpha \gamma \\ \beta \delta & 0\end{array}\right]$ and $BA=\left[\begin{array}{cc}0& \gamma \beta \\ \delta \alpha & 0\end{array}\right]$ Statement-1: $$\begin{array}{rl}& AB-BA=\left[\begin{array}{cc}0& \gamma (\alpha -\beta \\ \delta (\beta -\alpha )& 0\end{array}\right])\\ & |AB-BA|=(\alpha -{\beta }^2⟩\delta \ne 0\end{array}$$ $\therefore AB-BA$ is always an invertible matrix. Hence, statement $-1$ is true. But $AB-BA$ can be identity matrix if $\gamma =-\delta$ or $\delta =-\gamma$ So, statement $-2$ is false.