Definition and Properties of Determinants

The determinant is a scalar-valued function of square matrices that encodes fundamental information about linear transformations: invertibility, volume scaling, and orientation preservation.

DefinitionDeterminant

The determinant is a function $\det: M_n(\mathbb{F}) \to \mathbb{F}$ that assigns to each $n \times n$ matrix $A$ a scalar $\det(A)$ (also written $|A|$ ), defined recursively:

For $n = 1$ : $\det([a]) = a$

For $n \geq 2$ : Using cofactor expansion along row $i$ : $\det(A) = \sum_{j=1}^n (-1)^{i+j} a_{ij} \det(A_{ij})$

where $A_{ij}$ is the $(n-1) \times (n-1)$ matrix obtained by deleting row $i$ and column $j$ from $A$ .

Alternatively, the determinant can be characterized axiomatically as the unique function satisfying: (1) multilinearity in rows, (2) alternating property (swapping rows changes sign), and (3) $\det(I) = 1$ .

ExampleComputing Determinants

For $2 \times 2$ matrices: $\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$

For $3 \times 3$ matrices: $\det\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)$

Example: $\det\begin{bmatrix} 2 & 1 & 3 \\ 0 & -1 & 2 \\ 1 & 4 & 0 \end{bmatrix} = 2(-8) - 1(-2) + 3(1) = -16 + 2 + 3 = -11$

DefinitionProperties of Determinants

For $n \times n$ matrices $A$ and $B$ :

Multiplicativity: $\det(AB) = \det(A)\det(B)$
Transpose: $\det(A^T) = \det(A)$
Inverse: If $A$ is invertible, $\det(A^{-1}) = \frac{1}{\det(A)}$
Scalar multiplication: $\det(cA) = c^n\det(A)$
Row operations:
- Swapping rows multiplies determinant by $-1$
- Multiplying a row by $c$ multiplies determinant by $c$
- Adding a multiple of one row to another leaves determinant unchanged
Triangular matrices: $\det(A)$ equals the product of diagonal entries

Remark

The determinant has a geometric interpretation: for a linear transformation $T: \mathbb{R}^n \to \mathbb{R}^n$ with matrix $A$ , the value $|\det(A)|$ equals the factor by which $T$ scales $n$ -dimensional volume. The sign of $\det(A)$ indicates whether $T$ preserves or reverses orientation. Thus $\det(A) = 0$ means the transformation collapses volume, corresponding to non-invertibility.