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

The determinant \( |A| \) and the adjugate of \( A \) are crucial for assessing the consistency of a linear system \( AX = B \). 1. When \( |A| eq 0 \), matrix \( A \) is invertible, ensuring system consistency with the solution \( X = A^{-1} B \). 2. If \( |A| = 0 \), consistency is conditional. The system is inconsistent if \( \text{adj} B = 0 \). If \( \text{adj} B eq 0 \), infinitely many solutions exist. 3. A non-zero determinant \( |A| eq 0 \) guarantees a consistent system with a single, unique solution. Therefore, the system's consistency is uncertain when \( |A| = 0 \) and \( \text{adj} B = 0 \).