1. Tardigrade
  2. Question
  3. Mathematics
  4. Let A and B be real matrices of the form [α&0 0&β] and [0&γ δ&0], respectively. Statement 1: AB - BA is always an invertible matrix. Statement 2: AB-BA is never an identity matrix.

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

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

AIEEEAIEEE 2012Determinants

Solution:

Let A and B be real matrices such that
$A =\begin{bmatrix}\alpha&0\\ 0&\beta\end{bmatrix}$ and $B = \begin{bmatrix}0&\gamma\\ \delta&0\end{bmatrix}$
Now, $AB = \begin{bmatrix}0&\alpha\gamma \\ \beta\delta &0\end{bmatrix}$
and $BA =\begin{bmatrix}0&\gamma \beta\\ \delta\alpha &0\end{bmatrix}$
Statement-1:
$AB-BA = \begin{bmatrix}0&\gamma \left(\alpha-\beta\right) \\ \delta \left(\beta-\alpha\right) &0\end{bmatrix}$
$\left|AB-BA\right| = \left(\alpha-\beta^{2}\right)\delta \ne 0$
$\therefore AB - BA$ 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.