8.11 Determinant
The determinant of a square \(k\times k\) matrix \(\mathbf{A}\) is the scalar \[|\mathbf{A}| = \sum^k_{i=1}a_{1j}|\mathbf{A}_{1j}|(-1)^{1+j} \] where \(\mathbf{A}_{1j}\) is the \((k-1)\times(k-1)\) matrix obtained by deleting the first row and the \(j\)th column of \(\mathbf{A}\).
- Theorem: The determinant of a matrix \(\mathbf{A}\) is 0 if and only if \(\mathbf{A}\) is singular.