A) Statement 1 is true. Statement 2 is false.
B) Statement 1 is false, Statement 2 is true.
C) Statement 1 is true. Statement 2 is true; Statement 2 is a correct explanation of Statement!.
D) Statement 1 is true. Statement 2 is true, Statement 2 is not a correct explanation of Statement 1,
Correct Answer: A
Solution :
Let A and B be real matrices such that \[A=\left[ \begin{matrix} \alpha & 0 \\ 0 & \beta \\ \end{matrix} \right]\]and\[B=\left[ \begin{matrix} 0 & \gamma \\ \delta & 0 \\ \end{matrix} \right]\] Now,\[AB=\left[ \begin{matrix} 0 & \alpha \gamma \\ \beta \delta & 0 \\ \end{matrix} \right]\]and\[BA=\left[ \begin{matrix} 0 & \gamma \beta \\ \delta \alpha & 0 \\ \end{matrix} \right]\] Statement-1: \[AB-BA=\left[ \begin{matrix} 0 & \gamma \left( \alpha -\beta \right) \\ \delta \left( \beta -\alpha \right) & 0 \\ \end{matrix} \right]\] \[|AB-BA|={{\left( \alpha -\beta \right)}^{2}}\gamma \delta \ne 0\] \[\therefore \] 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.
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