If for a square matrix A, A.(adj A) = [{2025, 0, 0}, {0, 2025, 0}, {0, 0, 2025}], then the value of

Question

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

(A) \(1\)

(B) \(2025 + 1\)

(C) \((2025)^2 + 45\)

(D) \(2025 + (2025)^2\)

Answer 1

Correct option is (D) \(2025 + (2025)^2\)

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

So, \(A. (adj\ A) = \begin{bmatrix} 2025 &0&0\\0&2025&0\\0&0&2025\end{bmatrix} = 2025 I\)

\(\Rightarrow |A| = 2025\) & \(|adj\ A| = |A|^{3-1} = (2025)^2\)

\(\therefore |A| + |adj\ A| = 2025 + (2025)^2\)