Determinant¶

Definition¶

The determinant of a square matrix $\boldsymbol{A}=(a_{ij})$ of size $n\times n$ $n$ , denoted $\det(\boldsymbol{A})$ or $|\boldsymbol{A}|$ , is the scalar defined recursively by $\det(a)=a$ for a $1\times1$ matrix and, for $n\geq2$ $2$ ,

\det(\boldsymbol{A})=\sum_{j=1}^{n}(-1)^{1+j}a_{1j}\det(\boldsymbol{A}_{1j}),

where $\boldsymbol{A}_{1j}$ is the $(n-1)\times(n-1)$ matrix obtained from $\boldsymbol{A}$ by deleting row $1$ and column $j$ (cofactor expansion along the first row). It detects invertibility and encodes the volume scaling of the associated linear transformation.

Interpretation¶

The determinant condenses a square matrix into one number that answers two questions at once: by what factor the linear map $\boldsymbol{x}\mapsto\boldsymbol{A}\boldsymbol{x}$ scales areas (in $\mathbb{R}^2$ ) or volumes (in $\mathbb{R}^3$ ), and whether the map can be undone. A nonzero determinant means exactly that $\boldsymbol{A}$ is invertible.

The unit square spanned by e1 and e2, mapped by a matrix A onto a parallelogram spanned by Ae1 and Ae2 whose area 2.75 is the absolute value of the determinant.

The unit square of area $1$ is mapped onto the parallelogram spanned by the columns $\boldsymbol{A}\boldsymbol{e}_1$ and $\boldsymbol{A}\boldsymbol{e}_2$ of $\boldsymbol{A}=\begin{pmatrix}2&0.5\\0.5&1.5\end{pmatrix}$ $(2 0.5 0.5 1.5)$ . Its area is $|\det(\boldsymbol{A})|=|2\cdot1.5-0.5\cdot0.5|=2.75$ $2.75$ . A negative determinant would reverse the orientation, while a zero determinant would collapse the parallelogram to zero area and make the matrix singular.

The sign records orientation: swapping the two columns of $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ turns $ad-bc$ into $bc-ad$ $a d$ , the same area with the opposite sign. The product rule $\det(\boldsymbol{A}\boldsymbol{B})=\det(\boldsymbol{A})\det(\boldsymbol{B})$ fits the picture too: applying $\boldsymbol{B}$ and then $\boldsymbol{A}$ scales areas first by one factor and then by the other.

In practice determinants are computed by cofactor expansion along a convenient row or column, or by Sarrus' rule for $3\times3$ matrices. They give Cramer's rule for square systems, and the test whether a determinant is zero returns later in the course, when eigenvalues are computed.

A common slip is to treat the determinant as additive. In general $\det(\boldsymbol{A}+\boldsymbol{B})\neq\det(\boldsymbol{A})+\det(\boldsymbol{B})$ $det (B)$ : for $\boldsymbol{A}=\boldsymbol{B}=\boldsymbol{I}_2$ the left side is $\det(2\boldsymbol{I}_2)=4$ and the right side is $2$ . Likewise $\det(c\boldsymbol{A})=c^n\det(\boldsymbol{A})$ for an $n\times n$ matrix, not $c\det(\boldsymbol{A})$ .

Key Results¶

A square matrix is invertible if and only if $\det(\boldsymbol{A})\neq0$ $0$ .
$\det(\boldsymbol{A}\boldsymbol{B})=\det(\boldsymbol{A})\det(\boldsymbol{B})$ for square matrices of the same size.

Examples¶

For $\boldsymbol{A}=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ $(a c b d)$ , $\det(\boldsymbol{A})=ad-bc$ $b c$ .
$\det(\boldsymbol{I}_n)=1$ $1$ .

Prerequisites¶

Matrix

Sources¶

Math II: Linear Algebra · Week 4
Math V: Advanced Linear Algebra · Week 1 (revisited)

Aliases: det, matrix determinant Symbols: real numbers, sum

Last updated: 20 October 2026 03:02