If for a square matrix $A$, $A \cdot (\text{adj } A) = \begin{bmatrix} 2025 & 0 & 0 \\ 0 & 2025 & 0 \\ 0 & 0 & 2025 \end{bmatrix}$, then the value of $|A| + |\text{adj } A|$ is equal to

$2025 + (2025)^2$

The correct answer is Option (4) → $2025 + (2025)^2$ ##

For a square matrix $A$ of order $n \times n$, we have $A \cdot (\text{adj } A) = |A|I_n$, where $I_n$ is the identity matrix of order $n \times n$.

So, $A \cdot (\text{adj } A) = 2025 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = 2025 I_3$

$⇒|A| = 2025 \text{ and } n = 3$

$|\text{adj } A| = |A|^{3-1} = (2025)^2$

$|A| + |\text{adj } A|= 2025 + (2025)^2$