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:
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:
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For properties of adjoint matrices, always check if the matrix is scalar or symmetric.
Updated On: Jul 23, 2025
- \( 2a \)
- \( 2b \)
- \( 2c \)
- \( 0 \)
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The Correct Option is A
Solution and Explanation
Step 1: Condition for adjoint of \( A \)
If \( \text{adj} A = A \), then \( A \) must be a scalar multiple of the identity matrix. Let: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}. \] Step 2: Sum of elements
Since \( A = kI \) and the trace of \( A \) is \( 2a \), we find that: \[ a + b + c + d = 2a. \]
If \( \text{adj} A = A \), then \( A \) must be a scalar multiple of the identity matrix. Let: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}. \] Step 2: Sum of elements
Since \( A = kI \) and the trace of \( A \) is \( 2a \), we find that: \[ a + b + c + d = 2a. \]
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