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

For a \( 2 \times 2 \) matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), its adjugate matrix is defined as:

\[ \text{adj } A = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}. \]

The condition \( \text{adj } A = A \) implies:

\[ \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}. \]

Equating the corresponding elements yields:

\[ d = a, \quad -b = b, \quad -c = c, \quad a = d. \]

From \( -b = b \), it follows that \( b = 0 \). Similarly, from \( -c = c \), it follows that \( c = 0 \). Therefore, the matrix \( A \) takes the form:

\[ A = \begin{bmatrix} a & 0 \\ 0 & a \end{bmatrix}. \]

The sum of the elements of this matrix is:

\[ a + b + c + d = a + 0 + 0 + a = 2a. \]

Final Answer: \( \boxed{2a} \)