Let A be a 3 × 3 square matrix, such that A (adj A) = 2 I,
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,

$|a d j A|=|A|^{n-1}$

$\Rightarrow|a d j A|=2^{2}=4$