Question

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

• 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. Assertion is incorrect but Reason is correct

Answer 1

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

$A^{-1}=\frac{AdjA}{\left|A\right|}$

$\Rightarrow \left|A\right|I=A\cdot AdjA$

$\left|\left|A\right|I\right|=|A\cdot AdjA|$

$\Rightarrow {\left|A\right|}^n\left|I\right|=\left|A\right|\left|AdjA\right|$

$\Rightarrow \left|AdjA\right|={\left|A\right|}^{n-1}⟶1$

If $A=AdjA$

$\left|AdjAdj\right(A\left)\right|={\left|AdjA\right|}^{n-1}$

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

Again $AdjA=A$

$\Rightarrow \left|Adj\left(Adj\left(AdjA\right)\right)\right|={\left|AdjA\right|}^{{(n-1)}^2}$

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

$={\left|A\right|}^{{(n-1)}^3}⟶2$

From $1$ and $2$

Option C is correct