Matrix Inverse Using Minors, Cofactors, and Adjugate — Definition, Formula & Examples

Matrix inverse using minors, cofactors, and adjugate is a method that computes the inverse of a square matrix by finding the minor of each entry, applying a checkerboard of signs to get cofactors, transposing the cofactor matrix to form the adjugate, and dividing by the determinant.

For an invertible $n \times n$ matrix $A$ with $\det(A) \neq 0$ , the inverse is given by $A^{-1} = \frac{1}{\det(A)}\operatorname{adj}(A)$ , where $\operatorname{adj}(A)$ is the transpose of the cofactor matrix $C$ whose $(i,j)$ -entry is $C_{ij} = (-1)^{i+j} M_{ij}$ , and $M_{ij}$ is the minor obtained by deleting row $i$ and column $j$ from $A$ .

Key Formula

A^{-1} = \frac{1}{\det(A)}\operatorname{adj}(A) = \frac{1}{\det(A)}\, C^T

Where:

$A$ = The original invertible square matrix
$C$ = The cofactor matrix, with entries C_{ij} = (-1)^{i+j} M_{ij}
$C^T$ = Transpose of the cofactor matrix, also called the adjugate adj(A)
$\det(A)$ = Determinant of A (must be nonzero)
$M_{ij}$ = Minor: determinant of the submatrix after deleting row i and column j

How It Works

First, compute the minor

M_{ij}

for every entry by taking the determinant of the submatrix formed when you delete that entry's row and column. Next, build the cofactor matrix by multiplying each minor by

(-1)^{i+j}

, which flips the sign in a checkerboard pattern starting with

+

in the top-left. Then transpose the cofactor matrix — swap rows and columns — to get the adjugate. Finally, compute

\det(A)

; if it is nonzero, divide every entry of the adjugate by

\det(A)

to obtain

A^{-1}

Worked Example

Problem: Find the inverse of the matrix A = [[1, 2, 0], [0, 1, 1], [1, 0, 1]].

Determinant: Expand along the first row to find det(A).

\det(A) = 1(1\cdot1 - 1\cdot0) - 2(0\cdot1 - 1\cdot1) + 0 = 1(1) - 2(-1) = 3

Cofactor Matrix: Compute each cofactor C_{ij} = (-1)^{i+j} M_{ij}. The nine cofactors are:

C = \begin{bmatrix} +1 & +1 & -1 \\ -2 & +1 & +2 \\ +2 & -1 & +1 \end{bmatrix}

Adjugate (transpose of C): Transpose the cofactor matrix to get adj(A).

\operatorname{adj}(A) = C^T = \begin{bmatrix} 1 & -2 & 2 \\ 1 & 1 & -1 \\ -1 & 2 & 1 \end{bmatrix}

Divide by determinant: Divide every entry by det(A) = 3.

A^{-1} = \frac{1}{3}\begin{bmatrix} 1 & -2 & 2 \\ 1 & 1 & -1 \\ -1 & 2 & 1 \end{bmatrix}

Answer: A⁻¹ = (1/3) [[1, −2, 2], [1, 1, −1], [−1, 2, 1]]

Why It Matters

This method appears throughout linear algebra courses and is the standard way to derive Cramer's Rule. In engineering and physics, the adjugate formula gives a closed-form expression for the inverse, which is essential when working with symbolic matrices or proving theoretical results about determinants.

Common Mistakes

Mistake: Forgetting to transpose the cofactor matrix before dividing by the determinant.

Correction: The adjugate is the transpose of the cofactor matrix, not the cofactor matrix itself. Skipping the transpose swaps off-diagonal entries and produces a wrong result.