Question

If \[ B = \begin{bmatrix} 3 & \alpha & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3 \end{bmatrix} \] is the adjoint of a 3x3 matrix \( A \) and \( |A| = 4 \), then \( \alpha \) is equal to:

• A. 1
• B. 0
• C. -1
• D. -2

Hint:

The adjoint matrix \( B \) is the transpose of the cofactor matrix. The elements of \( B \) are cofactors corresponding to the elements of \( A \).

Answer 1

The Correct Option is A

Given that \( B \) is the adjoint of matrix \( A \) and \( |A| = 4 \). The relationship between the adjoint \( B \) and \( |A| \) is \( B = \text{adj}(A) = |A| \times A^{-1} \). The elements of \( B \) are the cofactors of the corresponding elements in \( A \). Specifically, \( B_{12} \) is the cofactor of \( A_{12} \), denoted as \( \alpha \). The cofactor \( \alpha \) is calculated as \( |A| \) multiplied by the minor of \( A_{12} \). We are given \( |A| = 4 \). The matrix \( B \) implies that the cofactor of \( A_{12} \) for \( A \) is 1. Conclusion: Thus, \( \alpha = 1 \).