The determinant packages several ideas into one scalar: invertibility, signed volume, orientation, and the effect of a linear map on top-dimensional area.
Alternating multilinear characterization
Theorem. There is a unique function \(\det:M_n(F)\to F\) such that, as a function of the columns, it is multilinear, alternating, and normalized by \(\det I=1\). It is given by \[ \det A=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)a_{\sigma(1),1}\cdots a_{\sigma(n),n}. \]
Proof. Multilinearity expands each column \(A_j=\sum_i a_{ij}e_i\). Alternation kills every term in which two selected basis vectors are equal. The surviving terms select a permutation \(\sigma\), and alternation assigns the sign \(\operatorname{sgn}(\sigma)\). This proves uniqueness and gives the formula; the formula itself is visibly multilinear, alternating, and normalized.
Row operations and invertibility
Proposition. Swapping two rows changes the sign of the determinant, multiplying a row by \(c\) multiplies the determinant by \(c\), and adding a multiple of one row to another leaves the determinant unchanged. A square matrix is invertible if and only if its determinant is nonzero.
Proof. The row operation rules follow from the same alternating multilinear properties applied to rows, or from \(\det A=\det A^\top\). Row reduction changes any matrix into an echelon matrix. If a zero row appears, the determinant is zero; otherwise the matrix reduces to the identity using operations whose determinant factors are nonzero, so the original determinant is nonzero.
Multiplicativity
Theorem. For square matrices \(A,B\in M_n(F)\), \[ \det(AB)=\det(A)\det(B). \]
Proof. Fix \(A\) and view \(B\mapsto\det(AB)\) as a function of the columns of \(B\). It is alternating and multilinear, because multiplication by \(A\) is linear on columns. By uniqueness of the determinant, it must be \(c\det B\). Evaluating at \(B=I\) gives \(c= \det A\).
Volume and orientation
Proposition. Over \(\mathbb R\), if the columns of \(A\) are \(v_1, \ldots,v_n\), then \(|\det A|\) is the \(n\)-dimensional volume of the parallelepiped spanned by those vectors. The sign records whether the ordered basis \((v_1, \ldots,v_n)\) has the same orientation as the standard basis.
A two-dimensional calculation
Calculation. For \[ A=\begin{pmatrix}2&1\\1&3\end{pmatrix}, \] the columns span a parallelogram with signed area \(\det A=2\cdot 3-1\cdot 1=5\). Reversing the two columns gives determinant \(-5\), so the magnitude records area while the sign records orientation.
Example. A shear \[ S=\begin{pmatrix}1&c\\0&1\end{pmatrix} \] has determinant \(1\). It changes angles and side lengths, but it preserves every parallelogram area. This is why adding a multiple of one row or column to another leaves the determinant unchanged.
Cofactors and Cramer's rule
Proposition. Expansion along row \(i\) gives \[ \det A=\sum_{j=1}^n(-1)^{i+j}a_{ij}\det A_{ij}, \] where \(A_{ij}\) deletes row \(i\) and column \(j\). If \(A\) is invertible and \(Ax=b\), then \[ x_j=\frac{\det A_j(b)}{\det A}, \] where \(A_j(b)\) replaces column \(j\) by \(b\).
Remark. Cramer's rule is conceptually important but usually computationally inefficient. Gaussian elimination is better for solving systems; determinants are better for structural statements about invertibility, volume, and characteristic polynomials.