Definition of Determinant

How a Matrix Changes Volume and Orientation

The determinant of a square matrix measures the $oriented volume scaling$ of its linear transformation. Its absolute value gives the volume factor, while its sign records whether orientation is preserved or reversed.

Every square matrix has one determinant. A zero determinant means that the image lies in a lower-dimensional set, so the transformation is not invertible.

Determinant is a special function that takes a square matrix and produces one real number:

\det : \mathbb{R}^{n \times n} \to \mathbb{R} : A \mapsto \det A

The sign and value of the determinant give three cases:

Determinant	Geometric effect	Invertibility
$\det A>0$	Scales volume by $\lvert\det A\rvert$ and preserves orientation	Invertible
$\det A<0$	Scales volume by $\lvert\det A\rvert$ and reverses orientation	Invertible
$\det A=0$	Maps the volume to zero and the image to a lower-dimensional set	Not invertible

The teal unit cube becomes the orange parallelepiped under the following matrix. The sloping edges show shear, but volume uses $base length and perpendicular heights$ .

A=\begin{pmatrix}2&1&0\\0&1&0\\0&0&1\end{pmatrix},\qquad \det A=2\cdot1\cdot1=2

Cube volume after stretching and shearing

The teal unit cube and orange image share the origin. Rotate the model to compare perpendicular heights with sloping edges.

$x$: $[0, 3]$
$y$: $[0, 1]$
$z$: $[0, 1]$
Lines: 24

The base length becomes 2, while both perpendicular heights remain 1. The volume therefore changes from 1 to 2. The sloping edge length is not a perpendicular height.

Three Determining Properties of Determinant

Only one determinant function satisfies the following $three properties$ . Each property fixes one behavior of that function: how it reacts to a row operation, what happens when two rows coincide, and which value the identity matrix must return. Together they determine the determinant completely.

Linear Property on Each Row

The determinant is $linear in each row while the other rows are fixed$ . This property is called multilinearity.

When we add two vectors in one row:

\begin{aligned} \det \begin{pmatrix} a_1 \\ \vdots \\ a_i + a_i' \\ \vdots \\ a_n \end{pmatrix} &= \det \begin{pmatrix} a_1 \\ \vdots \\ a_i \\ \vdots \\ a_n \end{pmatrix} \\ &\quad{}+ \det \begin{pmatrix} a_1 \\ \vdots \\ a_i' \\ \vdots \\ a_n \end{pmatrix} \end{aligned}

When we multiply one row by a scalar:

\det \begin{pmatrix} a_1 \\ \vdots \\ \lambda \cdot a_i \\ \vdots \\ a_n \end{pmatrix}

Alternating Property

The determinant is $alternating$ : if two rows are identical, its value is zero.

\det \begin{pmatrix} \vdots \\ a \\ \vdots \\ a \\ \vdots \end{pmatrix} = 0

Equivalently, swapping two rows reverses the sign of the determinant. Identical rows would be unchanged by the swap but would also have to change sign, so the only possible value is zero.

Normalization Property

Determinant is $normalized$ so that the determinant of the identity matrix is always $1$ :

\det I = 1

Here $I$ is the identity matrix of the relevant size. Multilinearity, alternation, and this normalization $determine the determinant uniquely$ .

Matrix Operations and Determinant

If we multiply the $entire matrix$ by a scalar $\lambda$ , the determinant gains one factor from each row:

\det(\lambda A) = \lambda^n \cdot \det A

This is because each row is multiplied by $\lambda$ , and there are $n$ total rows.

When we swap two rows of a matrix, the determinant changes sign:

\det B = -\det A

Adding a multiple of one row to a different row does not change the determinant:

\det \begin{pmatrix} \vdots \\ a_j \\ \vdots \\ a_i + \lambda a_j \\ \vdots \end{pmatrix}

Relationship with Matrix Invertibility

The determinant decides whether a square matrix $A$ is invertible. The following conditions are equivalent:

$Matrix A is invertible$
$There exists an inverse matrix$ $A^{-1}$ that satisfies $AA^{-1} = I$
$Full rank matrix$ : $\operatorname{rank}(A) = n$
$Trivial kernel$ : $\ker(A) = \{0\}$
$Columns are linearly independent$
$Rows are linearly independent$
$Determinant is not zero$ : $\det A \neq 0$

If the determinant is zero, the transformation maps space into a lower dimensional set, so it cannot be inverted.

If the rows are linearly dependent, one row is a linear combination of the others. Multilinearity expands the determinant into terms with repeated rows, and every such term is zero by the alternating property.

Multiplication Properties and Similar Matrices

The determinant of a matrix product equals the product of the two determinants:

\det(AB) = \det A \cdot \det B

If matrix $A$ is invertible, the determinant of its inverse is:

\det(A^{-1}) = \frac{1}{\det A}

Two matrices $A$ and $B$ are called $similar$ if there exists an invertible matrix $S$ such that:

B = S^{-1}AS

Similar matrices have the same determinant. Apply the product rule for determinants.

\det B = \det(S^{-1}AS)

If Gaussian elimination uses only row additions and $p$ row swaps, the determinant of the resulting upper triangular matrix $R$ is

\det R = (-1)^p \cdot \det A

If a row is also scaled, that scale factor must be tracked separately. With the allowed operations stated explicitly, triangularization provides a practical determinant algorithm.

Common Mistakes About Determinant

The determinant is $not additive in general$ . The following example disproves an addition rule.

As a simple example:

\det\left(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\right)

If the two determinants are calculated separately, the result is:

\det\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \det\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

Since $4 \neq 2$ , the determinant is not additive under matrix addition.