Question

Assertion :Let A be a $2\times 2$ matrix.
Statement - 1 : $\left|adj\right(adj\left(adjA\right)\left)\right|={\left|A\right|}^{{(n-1)}^3}$, where $n$ is order of matrix A. Reason: Statement - 2 : $\left|adjA\right|={\left|A\right|}^n$

• 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. Both Assertion and Reason are incorrect

Answer 1

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

Reason is incorrect because

$\left|adjA\right|={\left|A\right|}^{n-1}$

Assertion $\left|adj\left(adj\left(adjA\right)\right)\right|={\left|A\right|}^{{(n-1)}^3}$

$\left|adjA\right|={\left|A\right|}^{n-1}$

$adjA=detA{A}^{-1}$

$det\left(adjA\right)=det\left(det\left(A\right)A^{-1}\right)$

$det{A}^n\times \frac{1}{detA}=det{A}^{n-1}$

$A=AdjA$

$det\left(adj\left(adjA\right)\right)=det{\left(adjA\right)}^{n-1}$

$=det{\left(A\right)}^{{(n-1)}^2}$

Again $A=AdjA$

$=det\left(adj\left(adj\left(adjA\right)\right)\right)=det\left(adj{\left(A\right)}^{{(n-1)}^2}\right)$

$=det{\left(A\right)}^{{(n-1)}^3}$