Inverse Matrix Calculator: Cofactors & Adjoint Method for 2x2 and 3x3

Inverse Matrix Calculator: Cofactors & Adjoint Method (2x2 and 3x3)

Definitions: Cofactor Matrix and Adjoint

For a square matrix \( A \), the cofactor of element \( a_{ij} \) is:

\[ C_{ij} = (-1)^{i+j} \cdot M_{ij} \]

where \( M_{ij} \) is the minor (determinant of the submatrix after removing row i and column j).

The factor \((-1)^{i+j}\) determines the sign of the cofactor: positive if \(i+j\) is even, negative if \(i+j\) is odd.

The cofactor matrix \( \mathbf{C} \) is the matrix of all cofactors \( C_{ij} \).

The adjoint matrix (or adjugate) is the transpose of the cofactor matrix:

\[ \text{adj}(A) = \mathbf{C}^T \quad \text{so that} \quad [\text{adj}(A)]_{ij} = C_{ji} \]

The inverse matrix is then:

\[ A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) \]

Click on any cofactor value in the results to see the visual representation with the deleted row and column highlighted in red, and the remaining determinant calculated step by step. The sign factor is shown in red.

Matrix Inverse (Cofactor/Adjoint)

Choose 2x2 or 3x3. Enter fractions (like 2/3), decimals, or integers.

Results

det(A) =

Matrix of Cofactors (C) click any cell for visual details !

Adjoint Matrix (adj(A) = C^T)

Inverse Matrix (Exact fractions)

Step-by-Step: How the Inverse is Calculated

Step 1: Calculate the determinant of A. If zero, the matrix is singular and has no inverse.

Step 2: For each element, compute the minor (determinant of submatrix after deleting row i and column j) and apply the sign \((-1)^{i+j}\) to get the cofactor. The sign is positive (red) if i+j is even, negative if i+j is odd.

Step 3: Assemble the cofactor matrix C.

Step 4: Transpose C to obtain the adjoint matrix adj(A).

Step 5: Divide each element of adj(A) by det(A) to get the inverse.

Further Resources

Cofactor Calculation