Determinant

The determinant is a scalar-valued function of a square matrix, central to linear algebra. Written $\det(A)$ or $|A|$, it identifies whether a matrix is invertible and describes how its associated linear transformation scales volume. Unlike the matrix itself, the determinant is a single number. Its algebraic properties connect matrix multiplication, linear dependence, and the solution of linear equations; its geometric interpretation connects these operations with changes in area, volume, and orientation. (textbooks.math.gatech.edu)

Definition and elementary examples

For matrices over a field, such as the real numbers or complex numbers, the determinant can be characterized as the unique function satisfying three conditions: it is linear in each column separately, vanishes when two columns are identical, and equals $1$ on the identity matrix. Separate linearity makes it a multilinear map of the columns, not a linear function of the entire matrix. The equivalent definition using rows gives the same value. (textbooks.math.gatech.edu)

For a $1\times1$ matrix, the determinant is its sole entry. For a $2\times2$ matrix,

Thus,

An upper- or lower-triangular matrix has determinant equal to the product of its diagonal entries. This includes diagonal matrices, for which the formula is immediately visible. (textbooks.math.gatech.edu)

Algebraic properties

Elementary row operations affect determinants in precisely controlled ways:

• Interchanging two rows reverses the sign.
• Multiplying one row by a scalar $c$ multiplies the determinant by $c$.
• Adding a multiple of one row to another leaves the determinant unchanged.

The same rules hold for columns. In particular, identical rows, a zero row, or a row expressible as a linear combination of the others forces the determinant to vanish. Scaling every row of an $n\times n$ matrix gives $\det(cA)=c^n\det(A)$. (math.mit.edu)

For square matrices of the same size,

where $A^{\mathsf T}$ is the transpose. If an inverse matrix exists, multiplicativity yields

By contrast, determinants are generally not additive: $\det(A+B)$ need not equal $\det(A)+\det(B)$. Multilinearity applies only when the other rows or columns remain fixed. (textbooks.math.gatech.edu)

Invertibility and linear systems

Over a field, an $n\times n$ matrix is invertible exactly when its determinant is nonzero. This is equivalent to its columns having linear independence, its rank being $n$, and its null space containing only the zero vector. Geometrically, a zero determinant signals that the transformation loses at least one dimension. (textbooks.math.gatech.edu)

For a system of linear equations $Ax=b$, a nonzero determinant guarantees a unique solution for every right-hand side $b$. Cramer’s rule expresses its coordinates as

where $A_i$ replaces column $i$ of $A$ with $b$. If the determinant is zero, the system may instead have no solution or, over the real or complex numbers, infinitely many solutions; the right-hand side determines which occurs. (textbooks.math.gatech.edu)

Geometric interpretation

In geometry, the absolute determinant of a real matrix is the volume of the parallelepiped whose edges are its columns. In two dimensions this is parallelogram area; in three dimensions it is ordinary volume. Consequently, the linear transformation $x\mapsto Ax$ multiplies $n$-dimensional volume by $|\det(A)|$. (textbooks.math.gatech.edu)

For a nonsingular real matrix, a positive determinant preserves orientation and a negative determinant reverses it. A zero determinant collapses the space into a lower-dimensional image. For example, the diagonal matrix $\operatorname{diag}(2,3)$ expands area sixfold, whereas $\operatorname{diag}(-1,1)$ reflects the plane and has determinant $-1$. These interpretations explain why volume scaling factors multiply when transformations are composed. (textbooks.math.gatech.edu)

Expansion and computation

A cofactor combines a smaller determinant with a sign. If $A_{\widehat i,\widehat j}$ is obtained by deleting row $i$ and column $j$, define

Expansion along row $i$ then gives

Expansion along any column is equally valid. This recursive formula is useful for symbolic expressions and matrices containing many zeros, but unrestricted recursive expansion becomes expensive as matrix size increases. (textbooks.math.gatech.edu)

Gaussian elimination computes determinants more efficiently by reducing a matrix to triangular form while tracking row swaps and scaling. Standard dense elimination has computational complexity $O(n^3)$ in arithmetic operations. In numerical linear algebra, LU decomposition provides a practical implementation: the triangular factors supply diagonal products, while the row permutation supplies the sign. (textbooks.math.gatech.edu)

With floating-point arithmetic, diagonal products can overflow or underflow. An alternative stores the sign—or complex phase—separately from $\log|\det(A)|$, avoiding the need to represent the full product directly. (numpy.org)

Eigenvalues and calculus

The characteristic polynomial

is a degree-$n$ polynomial whose roots are the matrix’s eigenvalues, counted with algebraic multiplicity. Over the complex numbers, their product equals $\det(A)$, whether or not the matrix is diagonalizable. (textbooks.math.gatech.edu)

In multivariable calculus, the determinant of the Jacobian matrix describes local signed volume scaling for a differentiable coordinate transformation. Its absolute value supplies the volume correction in the change-of-variables formula for an integral, extending the linear volume interpretation to nonlinear mappings under the theorem’s regularity and invertibility conditions. (textbooks.math.gatech.edu)