Question

If \(A\) is an identity matrix of order \(n\), then \(A (\text{Adj } A)\) is a/an:

• A. identity matrix
• B. row matrix
• C. zero matrix
• D. skew symmetric matrix

Hint:

For any square matrix \(A\), if \(A\) is non-singular, then \(A \cdot \text{Adj}(A) = |A| \cdot I\). In particular, for the identity matrix, \(|A| = 1\) and \(\text{Adj}(A) = I\), so the product remains the identity matrix.

Answer 1

The Correct Option is A

We are given that \(A\) is an identity matrix of order \(n\). From matrix algebra, we know that: \[ A \cdot \text{Adj}(A) = |A| \cdot I \] For an identity matrix \(A = I\), we have: \[ |A| = |I| = 1 \quad \text{and} \quad \text{Adj}(I) = I \] So, \[ A \cdot \text{Adj}(A) = I \cdot I = I \] Hence, \[ A(\text{Adj }A) = I \] So, the result is again an identity matrix. \[ \boxed{A(\text{Adj }A) = I} \]