Cofactor Matrix - Signed Cofactors, Adjugate, and Inverse

A cofactor matrix calculator is a focused linear-algebra tool that takes a square 2x2 or 3x3 matrix A and lays out the full matrix of cofactors in the same shape as A, with each entry already multiplied by its (-1)^(i + j) sign factor. Drop in the four or nine entries of A, and the result panel shows the cofactor matrix C, the adjugate adj(A) = C^T, the determinant det(A), and the inverse A^-1 together.

• • Linear-algebra homework and exams: Verify the signed cofactors of a 2x2 or 3x3 matrix, then read the adjugate and the inverse from the same screen so the A^-1 = adj(A) / det(A) identity is checked in one pass.
• • Statics and small structural systems: Compute the inverse of a 3x3 Jacobian or stiffness matrix by hand, then check the cofactor pattern before trusting the inverse in a back-substitution.
• • Computer graphics and robotics: Invert a small rotation or scale matrix used in a shader, a calibration routine, or a kinematic chain, and read the cofactor matrix alongside the inverse to catch a sign error.
• • Statistics and econometrics: Cross-check the inverse of a 2x2 or 3x3 covariance matrix before feeding it into a regression or a Kalman update.

The sign factor starts at +1 in the top-left and flips at every step, which is the most common place a hand calculation goes wrong.

Because the adjugate is exactly the transpose of the cofactor matrix, Adjoint Matrix Calculator is the natural next step after this page once adj(A) is the only thing you still need.

The calculator reads the matrix size, collects the four 2x2 or nine 3x3 entries, and walks the cofactor definition in order: minor, sign factor, signed cofactor, full cofactor matrix, transpose to get the adjugate, and divide by the determinant to get the inverse.

• A: The input square matrix of size 2x2 or 3x3 with real entries.
• M_ij: The (i, j) minor: det of the (n - 1) x (n - 1) submatrix after deleting row i and column j.
• C_ij: The (i, j) cofactor: M_ij times the checkerboard sign factor (-1)^(i + j).
• (-1)^(i + j): Sign factor. +1 when i + j is even, -1 when odd.
• adj(A): The classical adjoint, equal to C^T.
• det(A): The determinant of A. The inverse is defined only when det(A) is not zero.

According to Wikipedia, the classical adjoint (also called the adjugate) of a square matrix A is the transpose of the cofactor matrix, defined as C_ij = (-1)^(i + j) times the minor M_ij.

According to MIT OpenCourseWare 18.06 (Strang), the formula A^-1 = adj(A) / det(A) is the cofactor expansion of the inverse and is the textbook alternative to row reduction for small square systems.

When the same cofactor C_ij is summed along a row or column to evaluate det(A) as a Laplace expansion, Cofactor Expansion Calculator walks the term-by-term calculation explicitly.

Four small ideas cover the entire pipeline from A to A^-1. They are the same concepts in any linear-algebra textbook.

When the same cofactors reappear inside the determinant of (A - lambda * I) for an eigenvalue problem, Characteristic Polynomial Calculator returns the characteristic polynomial without any hand sign bookkeeping.

Pick the matrix size, type the entries of A, and read the cofactor matrix, the adjugate, the determinant, and the inverse from the right-hand panel.

1. 1 Choose the matrix size: Select 2x2 to enter four entries, or 3x3 to enter nine entries. In 2x2 mode the bottom row and right column of the entry grid are ignored.
2. 2 Type the matrix entries: Real numbers, decimals allowed, negative numbers allowed. The calculation updates as you type.
3. 3 Read the cofactor matrix: The first result panel. Each cell is the matching (n - 1) x (n - 1) minor with the checkerboard sign already applied.
4. 4 Read the adjugate: The transpose of the cofactor matrix, equal to C^T. The off-diagonal of adj(A) is the off-diagonal of C swapped across the main diagonal.
5. 5 Read the determinant and the inverse: The determinant is the scalar that turns the adjugate into the inverse. The inverse panel shows adj(A) / det(A) when det(A) is not zero, or a singular-matrix warning.
6. 6 Reset or change the size: Reset restores the default 2x2 matrix; the size selector jumps to 3x3 without reloading the page.

Example: a student is asked for the cofactor matrix of A = [[1, 2], [3, 4]]. They pick 2x2, type the four entries, and read [[4, -3], [-2, 1]] in the first result panel. The adjugate shows [[4, -2], [-3, 1]], the determinant reads -2, and the inverse reads [[-2, 1], [1.5, -0.5]].

For a 3x3 problem where the cofactors are easier to remember as determinants of 2x2 cross-product slices, Cross Product Calculator returns the same 2x2 determinant directly from two vector inputs.

The cofactor definition is short, but going from a 3x3 matrix to the inverse by hand is a multi-step bookkeeping task. The calculator keeps the whole pipeline on screen at once.

• • Cofactor matrix shown as its own step: The cofactor matrix sits between the input and the adjugate, so the user can audit the checkerboard sign pattern before the transpose.
• • 2x2 and 3x3 in one tool: A size selector swaps between the four-entry 2x2 input and the nine-entry 3x3 input.
• • Inverse through the adjugate identity: The inverse is computed as adj(A) / det(A), the textbook alternative to Gaussian elimination for small square matrices.
• • Singular-matrix warning built in: When the determinant is exactly zero, the inverse panel switches to a warning. C and adj(A) are still returned.
• • Decimal-friendly arithmetic: Entries can be decimals like 1.5 or 0.25, with four decimal places of precision in the result panels.

The result panel is laid out so the identity A * adj(A) = det(A) * I is visible: C in the first panel, adj(A) in the second, det(A) as the scalar that ties them together, and A^-1 in the last panel.

For a symmetric positive-definite matrix where the inverse could also be built from a Cholesky factor, Cholesky Decomposition Calculator returns L and L^T so the same matrix can be inverted by forward and back substitution.

A handful of input choices and structural facts decide whether the cofactor matrix lines up with the matrix you intended, and whether the inverse is meaningful at all.

• • The calculator is restricted to 2x2 and 3x3 square matrices. For 4x4 and larger, a row-reduction approach is more practical.
• • The cofactor definition extends to complex entries, but this calculator is built for real 2x2 and 3x3 inputs.
• • Results are rounded to four decimal places for display. A clean fraction like 1/3 will read as 0.3333 on the screen.

The same two ideas cover everything: the sign convention is fixed, and the determinant decides whether the inverse is defined.

According to Wolfram MathWorld, the cofactor C_ij of an n x n matrix A is the determinant of the (n - 1) x (n - 1) submatrix obtained by deleting row i and column j, multiplied by the sign factor (-1)^(i + j).

To go from a cofactor matrix to the solution vector of A * x = b, System of Equations Calculator returns x for 2x2 and 3x3 systems and uses row reduction when det(A) is zero.