Question

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

Hint:

Remember the adjoint matrix properties: adjoint \( \times \) determinant gives the original matrix.

Answer 1

We are given: \( B = \begin{bmatrix} 3 & a & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3 \end{bmatrix} \)

Step 1: Relationship between adjoint and determinant The adjoint of a matrix \( A \), denoted as \(\text{adjoint}(A)\), satisfies the following property: \[ A \cdot \text{adjoint}(A) = |A| \cdot I \] where \( I \) is the identity matrix and \( |A| \) is the determinant of \( A \). Given \( |A| = 4 \), we have: \[ A \cdot B = 4 \cdot I \] 

Step 2: Properties of the adjoint matrix The adjoint matrix is the transpose of the cofactor matrix of \( A \). For the adjoint to be valid, the entries in \( B \) must satisfy this relationship when multiplied with \( A \). 

Step 3: Expand \( \text{adjoint}(A) \) for consistency Matrix \( B \) is symmetric, so it represents the adjoint matrix. For the adjoint to hold, the diagonal entries of \( B \) must match the cofactors of \( A \), and off-diagonal entries must not affect the determinant calculation adversely. The symmetry of \( B \) suggests \( a = 1 \) ensures consistency with the determinant \( |A| = 4 \). 

Conclusion: Thus, the value of \( a \) is: \[ \boxed{1} \] ---