The determinant is a scalar that captures key information about a square matrix. For the 2×2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) it is simply \(ad - bc\). Geometrically, the determinant measures how the associated linear map scales area (in two dimensions) or volume (in higher dimensions); a zero determinant signals that the matrix is singular — its columns are linearly dependent and the map collapses dimension.
Determinants obey several useful identities. They are multiplicative, \(\det(AB) = \det(A)\det(B)\), and invariant under transposition, \(\det(A^T) = \det(A)\). The determinant of an inverse is \(\det(A^{-1}) = 1/\det(A)\). Swapping two rows flips the sign of the determinant, and for a triangular matrix the determinant is simply the product of its diagonal entries. These properties make \(\det(A) = 0\) a clean test for non-invertibility and let us derive many determinant formulas from a few basic cases.
For larger matrices, determinants are computed by cofactor expansion: \(\det(A) = \sum_j (-1)^{i+j} a_{ij} M_{ij}\) along any fixed row \(i\), where \(M_{ij}\) is the \((i,j)\) minor, namely the determinant of the submatrix obtained by deleting row \(i\) and column \(j\). Cramer's rule then expresses each component of the solution to \(Ax = b\) as a ratio \(x_i = \det(A_i)/\det(A)\), where \(A_i\) is the matrix obtained by replacing the \(i\)-th column of \(A\) with \(b\). The Cayley–Binet formula generalizes a piece of this: for \(A\) of size \(m \times n\) and \(B\) of size \(n \times m\), \(\det(AB)\) equals the sum of the determinants of all \(n \times n\) submatrices formed by choosing \(n\) columns of \(A\) together with the corresponding rows of \(B\). A related differentiation rule concerns time-dependent matrices: \(\frac{d}{dt} \det(A(t)) = \det(A) \cdot \mathrm{tr}(A^{-1} A')\), tying the rate of change of the determinant to the trace.