Home Statistical Dictionary Determinant

Determinant

A single number computed from a square matrix that indicates whether the matrix is invertible (nonzero determinant) and measures how much the matrix scales volume -- the quantity appearing in the multivariate normal density's normalizing constant.

Linear Algebra for StatisticsIntermediateFree

In Plain English

The determinant boils an entire square matrix down to one number that answers two closely related questions at once: can this matrix be inverted (yes, exactly when the determinant isn't zero), and how much does this matrix stretch or shrink space (the determinant's absolute value is that scaling factor for volume). A zero determinant is a red flag in statistics, it means the matrix is singular, and a covariance matrix with a zero determinant is exactly the degenerate case where the multivariate normal density formula breaks down.

Definition

The determinant of a square matrix $A$ , denoted $det (A)$ or $∣ A ∣$ , is a scalar computed from the matrix's entries that is nonzero exactly when $A$ is invertible; for a $2 \times 2$ matrix, $det (a c b d) = a d - b c$ , and larger matrices are computed via cofactor expansion or, more efficiently in practice, via a matrix decomposition such as LU. Geometrically, $∣ det (A) ∣$ is the factor by which the linear transformation represented by $A$ scales volume, and the determinant equals the product of the matrix's eigenvalues, $det (A) = \prod_{i} λ_{i}$ , which is why a matrix with any zero eigenvalue has a zero determinant and is singular. In statistics, the determinant of a covariance matrix appears directly in the multivariate normal density's normalizing constant, $(2 π)^{- k /2} ∣ Σ ∣^{- 1/2}$ , and its logarithm, the generalized variance, is sometimes used as a single scalar summary of a multivariate spread.

Formula

det (a c b d) = a d - b c

Notation

det (A) or ∣ A ∣

—the determinant of A

λ_{i}

—the i-th eigenvalue of A

Worked Example

For $A = (4638)$ , the determinant is $det (A) = (4) (8) - (3) (6) = 32 - 18 = 14$ , a nonzero value confirming $A$ is invertible; a matrix like $(2142)$ , whose second row is exactly half the first, has $det = (2) (2) - (4) (1) = 0$ , correctly flagging it as singular (its rows are linearly dependent).

Properties

A zero determinant is equivalent to a singular (non-invertible) matrix, which is equivalent to the matrix having at least one zero eigenvalue and less than full rank -- these three conditions (zero determinant, singularity, rank deficiency) are different ways of describing exactly the same underlying situation, and any one of them implies the other two.
The determinant of a product of matrices equals the product of their individual determinants, $det (AB) = det (A) det (B)$ , a multiplicative property with no analog for addition, $det (A + B) \neq = det (A) + det (B)$ in general.
The logarithm of a covariance matrix's determinant (log generalized variance) is used as a single scalar summary of overall multivariate spread across several variables at once, and appears directly in the log-likelihood of the multivariate normal distribution, making it a quantity that shows up throughout maximum likelihood estimation for multivariate models.

At a Glance

2x2 formuladet(A) = ad - bc

Nonzero determinant meansA is invertible (nonsingular), full rank

Equalsthe product of A's eigenvalues

In statistics, appears inthe multivariate normal density's normalizing constant

Last updated September 8, 2026← Back to the Dictionary