Question:
Let $A$ be a $3 \times 3$ square matrix, such that $A(\operatorname{adj} A)=2 I$, where $I$ is the identity matrix. Write the value of $|\operatorname{adj} A|$.
Solution:
We know that for a matix $A$ of order $n, A \cdot(\operatorname{adj} A)=|A| I_{n}$, where $I$ is the identity matrix.
Given: $A \cdot(\operatorname{adj} A)=2 I$
$\Rightarrow|A| I=2 I$
$\Rightarrow|A|=2$
Now,
$|a d j A|=|A|^{n-1}$
$\Rightarrow|a d j A|=2^{2}=4$
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