Let A and B be real matrices of the form [ * 20 c &0 0& ] and [ *… - Vidyadip

MCQ

Let $A$ and $B$ be real matrices of the form $\left[ {\begin{array}{*{20}{c}}
\alpha &0\\
0&\beta
\end{array}} \right]$ and $\left[ {\begin{array}{*{20}{c}}
0&\gamma \\
\delta &0
\end{array}} \right]$, respectively

Statement $1$ : $AB - BA$ is always an invertible matrix

Statement $2$ : $AB -BA$ is never an identity matrix

✓
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 $1$.
D
Statement $1$ is true, Statement $2$ is true,Statement $2$ is not a correct explanation of Statement $1$.

✓

Answer

Correct option: A.

Statement $1$ is true, Statement $2$ is false

a
Let $A$ and $B$ be real matrices such that

$A = \left[ {\begin{array}{*{20}{c}}
\alpha &0\\
0&\beta
\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}
0&\lambda \\
\delta &0
\end{array}} \right]$

Now, $AB = \left[ {\begin{array}{*{20}{c}}
0&{\alpha \gamma }\\
{\beta \delta }&0
\end{array}} \right]$

and $BA = \left[ {\begin{array}{*{20}{c}}
0&{\gamma \beta }\\
{\delta \alpha }&0
\end{array}} \right]$

Statement - $1$:

$AB - BA = \left[ {\begin{array}{*{20}{c}}
0&{\gamma \left( {\alpha - \beta } \right)}\\
{\delta \left( {\beta - \alpha } \right)}&0
\end{array}} \right]$

$\left| {AB - BA} \right| = {\left( {\alpha - \beta } \right)^2}\delta \ne 0$

$\therefore AB - BA$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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