Determinant

Every linear map stretches, compresses, rotates, or reflects the space it acts on. The determinant captures the net effect in a single number: it is the signed factor by which the map scales volumes. If the determinant is 2, the map doubles every volume; if it is negative, the map also flips orientation; if it is 0, the map collapses space entirely — and the map becomes uninvertible.

The 2×2 case

For a 2×2 matrix

$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix},$

the determinant is

\det(A) = ad - bc. \tag{1}

The geometric meaning is immediate: the two columns of $A$ are vectors $\mathbf{u} = (a, c)^\top$ and $\mathbf{v} = (b, d)^\top$ in $\mathbb{R}^2$. The magnitude $|\det(A)| = |ad - bc|$ equals the area of the parallelogram spanned by $\mathbf{u}$ and $\mathbf{v}$.

The sign carries orientation information. If the columns go counterclockwise (standard orientation), $\det(A) > 0$; if they go clockwise (reversed orientation), $\det(A) < 0$; and $\det(A) = 0$ when the columns are linearly dependent — the parallelogram collapses to a line segment with zero area.

The 3×3 case

Write

$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}.$

The determinant is

$$\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}). \tag{2}$$

Each parenthesized expression is itself a 2×2 determinant — the submatrix you get by deleting row 1 and the column of the corresponding entry. This “expansion along the first row” is the starting point for the general definition.

The general definition

For an $n \times n$ matrix $A$ over a field $F$, the determinant is defined recursively by cofactor expansion along row 1:

\det(A) \coloneqq \sum_{j=1}^{n} (-1)^{1+j}\, a_{1j}\, \det(A_{1j}), \tag{3}

where $A_{ij}$ denotes the $(i,j)$ minor — the $(n-1) \times (n-1)$ submatrix obtained by deleting row $i$ and column $j$ from $A$. The scalar

$C_{ij} \coloneqq (-1)^{i+j} \det(A_{ij})$

is called the $(i,j)$ cofactor of $A$. The base case is $\det([a]) = a$ for a $1 \times 1$ matrix.

One of the key theorems, proved in Properties of the Determinant, is that expanding along any row or column gives the same value. This is far from obvious from the definition above but makes computing determinants much more flexible in practice.

Geometric interpretation

In $\mathbb{R}^n$, $|\det(A)|$ is the $n$-dimensional volume of the parallelepiped spanned by the column vectors of $A$. In $\mathbb{R}^2$ this is area; in $\mathbb{R}^3$ it is volume in the usual sense.

The sign indicates orientation: $\det(A) > 0$ means the columns form a positively oriented basis (same handedness as the standard basis); $\det(A) < 0$ means orientation is reversed.

Because $\det(A)$ is the volume-scaling factor of the linear map $T_A(x) = Ax$, you have

$\operatorname{Vol}(T_A(S)) = |\det(A)| \cdot \operatorname{Vol}(S)$

for any bounded region $S \subseteq \mathbb{R}^n$. This identity is the foundation of the change-of-variables formula in multivariable integration.

Determinant and invertibility

The most important single fact about determinants:

A \text{ is invertible} \iff \det(A) \ne 0. \tag{4}

If $\det(A) = 0$, the map $T_A$ collapses at least one dimension — volume becomes zero, so the map cannot be reversed. If $\det(A) \ne 0$, the map scales volumes by a nonzero factor and is therefore bijective, hence invertible. An explicit formula for $A^{-1}$ in terms of determinants is developed in Properties of the Determinant.

Summary

• The determinant $\det(A)$ is a scalar assigned to every square matrix, defined recursively by cofactor expansion: $\det(A) = \sum_{j=1}^n (-1)^{1+j} a_{1j} \det(A_{1j})$.
• For $n = 1$: $\det([a]) = a$. For $n = 2$: $\det\!\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc$.
• Geometrically, $|\det(A)|$ is the $n$-dimensional volume of the parallelepiped spanned by the columns of $A$; the sign records orientation.
• The linear map $T_A$ scales all volumes by $|\det(A)|$.
• $A$ is invertible if and only if $\det(A) \ne 0$.
• Practical computation and further properties — expansion along any row/column, row-operation rules, multiplicativity, the inverse formula, and Cramer’s rule — are covered in Properties of the Determinant.