Determinants

A scalar associated with a square matrix measuring the signed volume scaling factor of the linear transformation it encodes.

Diagram · Determinant as Area

Determinant as signed area scaling

What det measures

The columns of A make a parallelogram. Its signed area is the determinant.

det(A) = 2.75

area scale = 2.75

a=2.0b=0.5c=0.5d=1.5

Definition

The determinant of a square matrix $A$ , written $\det(A)$ or $|A|$ , is a scalar that captures essential information about the matrix.

For a $2\times2$ matrix:

$\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc$

Geometric meaning: $|\det(A)|$ is the area of the parallelogram formed by the columns of $A$ (in 2D) or the volume of the parallelepiped (in 3D). The sign indicates whether the transformation preserves or reverses orientation.

$\det(A) = 0$ : the matrix is singular (not invertible, columns are linearly dependent)
$\det(A) \neq 0$ : the matrix is invertible

2×2 determinant

$A = \begin{pmatrix}3&1\\2&4\end{pmatrix}$ : $\det(A) = 3\cdot4 - 1\cdot2 = 12 - 2 = 10$ .

The transformation $A$ scales areas by a factor of 10. Since $\det(A) > 0$ , orientation is preserved.

Try it

Find $\det\begin{pmatrix}2&-1\\4&-2\end{pmatrix}$ . What does this tell you about the matrix?

Solution

$\det = 2(-2) - (-1)(4) = -4 + 4 = 0$ .

The matrix is singular — not invertible. Indeed, row 2 = $2 \times$ row 1, so the rows are linearly dependent. The columns also span only a 1D subspace.

Related concepts

Needs first

Matrices Matrix Multiplication

Eigenvalues and Eigenvectors Linear Transformations Matrix Inverse

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