Question

Let \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] 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:

The adjugate matrix of a diagonal matrix is the matrix itself. Use properties of adjugates for simplifications.

Answer 1

The Correct Option is A

Let \( A \) be a \( 2 \times 2 \) matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \). The adjugate of \( A \) is given by \( \text{adj } A = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \). If \( \text{adj } A = A \), then \( \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \). Equating the corresponding elements yields \( d = a \), \( -b = b \), \( -c = c \), and \( a = d \). The equations \( -b = b \) and \( -c = c \) imply that \( b = 0 \) and \( c = 0 \), respectively. Therefore, the matrix \( A \) must be of the form \( A = \begin{bmatrix} a & 0 \\ 0 & a \end{bmatrix} \). The sum of the elements of \( A \) is \( a + b + c + d = a + 0 + 0 + a = 2a \).Final Answer: \( \boxed{2a} \)