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Matrix Inverse

The matrix that, multiplied by the original square matrix, produces the identity matrix -- the matrix-algebra analog of dividing by a number, and the operation solving the ordinary least squares normal equations for the fitted regression coefficients.

Linear Algebra for StatisticsIntermediateFree

In Plain English

The matrix inverse plays the role that division plays for ordinary numbers, matrix algebra has no direct division operation, so 'undoing' a matrix multiplication means multiplying by its inverse instead. This is exactly the step that turns the abstract regression equation X'Xβ = X'y into an explicit formula for the coefficients, β-hat = (X'X)^-1 X'y, multiplying both sides by the inverse of X'X isolates β the same way dividing both sides of 3x=6 by 3 isolates x.

Definition

For a square matrix $A$ , its inverse $A^{- 1}$ , if it exists, is the unique matrix satisfying $A A^{- 1} = A^{- 1} A = I$ . A square matrix is invertible (nonsingular) exactly when its determinant is nonzero, equivalently when it has full rank, and a non-invertible (singular) matrix, one with zero determinant, does not admit a true inverse, requiring a generalized inverse instead when a matrix-algebra formula still needs to be evaluated. The matrix inverse is what turns the ordinary least squares normal equations $X^{⊤} X \hat{β} = X^{⊤} y$ into the explicit closed-form solution $\hat{β} = (X^{⊤} X)^{- 1} X^{⊤} y$ , and inverse covariance matrices (precision matrices) appear directly in the multivariate normal density and in Mahalanobis distance.

Formula

A A^{- 1} = A^{- 1} A = I

Notation

A^{- 1}

—the inverse of A

Worked Example

For the 2x2 matrix $A = (2111)$ , the determinant is $2 (1) - 1 (1) = 1$ , so the inverse exists and equals $A^{- 1} = \frac{1}{1} (1 - 1 - 1 2) = (1 - 1 - 1 2)$ . Multiplying $A A^{- 1}$ confirms the identity matrix results: the top-left entry is $2 (1) + 1 (- 1) = 1$ , and working through the rest gives exactly $I_{2}$ .

Properties

A matrix inverse exists only for square matrices with nonzero determinant -- a non-square matrix has no true inverse at all (though it may have a one-sided or generalized inverse), and a square matrix with zero determinant (a singular matrix) also has no inverse, no matter how close to nonsingular it may appear numerically.
The inverse of a matrix product reverses order, $(AB)^{- 1} = B^{- 1} A^{- 1}$ , mirroring the same order-reversal rule that applies to the transpose of a product -- a detail that must be tracked carefully when algebraically manipulating matrix equations that involve several inverted products.
In practice, statistical software rarely computes a matrix inverse explicitly to solve a linear system, numerically stable factorization-based methods (like QR or Cholesky decomposition) solve the same equations more efficiently and with less accumulated rounding error than forming $(X^{⊤} X)^{- 1}$ directly, even though the inverse remains the standard way to write the solution algebraically.

At a Glance

Defining propertyA A^-1 = A^-1 A = I

Exists exactly whenA is square with nonzero determinant (full rank)

If it doesn't existA is singular -- use a generalized inverse instead

Solvesthe OLS normal equations for beta-hat = (X'X)^-1 X'y

Last updated September 8, 2026← Back to the Dictionary