Question

Assertion :There are only finitely many $2\times 2$ matrices which commute with the matrix $\left[\begin{array}{cc}1& 2\\ -1& -1\end{array}\right]$.
Reason: If A is non-singular, then it commutes with I, adj A and $A^{-1}$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

Assertion:

$\left[\begin{array}{cc}1& 2\\ -1& -1\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}1& 2\\ -1& -1\end{array}\right]$
$\left[\begin{array}{cc}a+2c& b+2d\\ -a-c& -b-d\end{array}\right]=\left[\begin{array}{cc}a-b& 2a-b\\ c-d& 2c-d\end{array}\right]$
$\Rightarrow 2c=-b$ & $b=a-d$

Thus assertion is incorrect.
$\therefore$ infinite matrix are there.
Reason:
$AI=IA\Rightarrow A\left(adjA\right)=\left(adjA\right)A$
$\Rightarrow A{A}^{-1}=A^{-1}A$

Thus reason is correct.