Key Concept: Condition for a Unique Solution

For a system of linear equations to have a unique solution, the determinant of its coefficient matrix must be non-zero. This is a fundamental concept in linear algebra, often associated with Cramer's Rule or the invertibility of the coefficient matrix. Consider a system of $n$ linear equations in $n$ variables, represented in matrix form as $AX = B$. Here, $A$ is the $n \times n$ coefficient matrix, $X$ is the column matrix of variables, and $B$ is the column matrix of constants. The system has a unique solution if and only if $\det(A) \ne 0$. If $\det(A) = 0$, the system either has no solution or infinitely many