Question

Let A and B be real matrices of the form $\left[\begin{array}{cc}\alpha & 0\\ 0& \beta \end{array}\right]$ and $\left[\begin{array}{cc}0& \gamma \\ \delta & 0\end{array}\right]$, respectively.
Statement 1: AB - BA is always an invertible matrix.
Statement 2: AB-BA is never an identity matrix.

• A. Statement 1 is true. Statement 2 is false
• B. Statement 1 is folse. Statement 2 is true
• C. Statement 1 is true. Statement 2 is true; Statement 2 is a correct explanation of Statement 1
• D. Statement 1 is true. Statement 2 is true, Statement 2 is not a correct explanation of Statement 1

Answer 1

Answer: (A)

Let A and B be real matrices such that
$A=\left[\begin{array}{cc}\alpha & 0\\ 0& \beta \end{array}\right]$ and $B=\left[\begin{array}{cc}0& \gamma \\ \delta & 0\end{array}\right]$
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:
$AB-BA=\left[\begin{array}{cc}0& \gamma \left(\alpha -\beta \right)\\ \delta \left(\beta -\alpha \right)& 0\end{array}\right]$
$\left|AB-BA\right|=\left(\alpha -{\beta }^2\right)\delta \ne 0$
$\therefore AB-BA$ 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.