Question

Let \( \Delta = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) be a square matrix such that \( \text{adj} A = A \). Then, \( (a + b + c + d) \) is equal to:

• A. \( 2a \)
• B. \( 2b \)
• C. \( 2c \)
• D. \( 0 \)

Hint:

For properties of adjoint matrices, always check if the matrix is scalar or symmetric.

Answer 1

The Correct Option is A

Step 1: Adjoint condition for \( A \)
If \( \text{adj} A = A \), then \( A \) must be of the form \( kI \), where \( k \) is a scalar and \( I \) is the identity matrix. Consider: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}. \] Step 2: Sum of entries
Given \( A = kI \) and \( \text{Tr}(A) = a+d \), we have: \[ a + b + c + d = \text{Tr}(A). \]