If the determinant is non zero, then it means that the system of equations represented by its matrix has a unique solution. If the determinant is zero, then it must mean that the system is a) inconsistent, or b) has infinitely many solutions. Which is it, a) or b)?
2026-03-30 03:53:47.1774842827
What does it mean if the determinant equals $0$?
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The simplest matrix with determinant $0$ is the $1\times 1$ matrix $[0]$. It represents equations of the form $$ 0x = b $$ for some real number $b$. Think of a real number $b$ that makes the system inconsistent, and a real number $b$ which makes the system have infinitely many solutions.
So if the coefficient matrix of a system of equations has determinant $0$, then the system is either inconsistent, or it has infinitely many solutions. You can't tell which one it is just from the matrix, though.