Question

If \( A = \begin{bmatrix} a & 1 & 2 \\ 1 & b & 3 \\ c & 1 & 3 \end{bmatrix} \) and \( \text{Adj } A = \begin{bmatrix} 7 & -1 & -5 \\ -3 & 9 & 5 \\ 1 & -3 & 5 \end{bmatrix} \), then \( a^2 + b^2 + c^2 = \) ?

• A. \(10\)
• B. \(14\)
• C. \(11\)
• D. \(29\)

Hint:

For adjoint-based questions, remember the key property: \( A \cdot \text{Adj } A = |A| I \).

Answer 1

The Correct Option is A

Step 1: Use the Property of Adjugate Matrix For a given \( n \times n \) matrix \( A \), the relation between \( A \) and its adjugate is: \[ A \cdot \text{Adj } A = \text{det}(A) \cdot I \] where \( I \) is the identity matrix. This means: \[ A \cdot \text{Adj } A = \text{det}(A) \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
Step 2: Multiply the Given Matrices Multiplying \( A \) with \( \text{Adj } A \): \[ \begin{bmatrix} a & 1 & 2 \\ 1 & b & 3 \\ c & 1 & 3 \end{bmatrix} \begin{bmatrix} 7 & -1 & -5 \\ -3 & 9 & 5 \\ 1 & -3 & 5 \end{bmatrix} \] Performing matrix multiplication: \[ = \begin{bmatrix} 7a - 3 + 2 & -a + 9 - 6 & -5a + 5 + 10 \\ 7 - 3b + 3 & -1 + 9b - 9 & -5 + 5b + 15 \\ 7c & -c & -5c + 20 \end{bmatrix} \]
Step 3: Compare with Determinant Condition Since: \[ A \cdot \text{Adj } A = \det(A) I \] Comparing with \( \det(A) I \), we obtain: \[ a^2 + b^2 + c^2 = 10. \] Thus, the correct answer is: \[ \boxed{10} \]