Question

A system of linear equations is represented as \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix. Then the system of equations is:

Hint:

If \( |A| \neq 0 \), the system has a unique solution. If \( |A| = 0 \), the system may have no solution or infinitely many solutions, depending on the value of \( \text{adj} B \).

Answer 1

For a system of linear equations represented as \( AX = B \), the determinant \( |A| \) and the adjugate of \( A \) play a key role in determining the consistency of the system. 1. If \( |A| \neq 0 \), the matrix \( A \) is invertible, and the system is consistent. The solution is given by: \[ X = A^{-1} B \] 2. If \( |A| = 0 \), the system may be inconsistent or have infinitely many solutions, depending on whether \( \text{adj} B = 0 \). If \( \text{adj} B = 0 \), the system has no solution (inconsistent); otherwise, there may be infinitely many solutions. 3. If \( |A| \neq 0 \), the system is always consistent with a unique solution. Thus, the correct answer is (D) May or may not be consistent if \( |A| = 0 \) and \( \text{adj} B = 0 \).