Adjoint Matrix Calculator - Cofactor and Inverse Steps

An adjoint matrix calculator returns the classical adjoint (also called the adjugate) of a square matrix along with the cofactor matrix, the determinant, and the inverse in one pass. The adjugate bridges the determinant and the matrix inverse through the identity A * adj(A) = det(A) * I.

• • Linear algebra homework and exams: Verify cofactor expansions and the A^-1 = adj(A) / det(A) identity on 2x2 and 3x3 problems.
• • Statics and small structural systems: Solve 3x3 equilibrium problems by hand, then check the inverse you used to back out the internal forces.
• • Computer graphics and robotics: Compute the inverse of a small rotation or scale matrix to undo a transform in a shader, calibration routine, or kinematic chain.
• • Statistics and econometrics: Cross-check the inverse of a 2x2 or 3x3 covariance matrix you wrote down on paper before feeding it into a regression.

Many textbooks shorten 'classical adjoint' to just 'adjoint'. Each cofactor is the matching minor of A multiplied by (-1)^(i + j), and the adjugate is the transpose of the cofactor matrix.

For solving 2x2 and 3x3 linear systems by Cramer's rule, which is built on the same cofactor and determinant backbone, the system of equations calculator is the natural companion.

The calculator reads the matrix size, collects the four 2x2 or nine 3x3 entries, and walks the cofactor expansion in order: minor, signed cofactor, transpose to get the adjugate, then 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: the determinant of the (n - 1) x (n - 1) submatrix of A after removing row i and column j.
• C_ij: the (i, j) cofactor: the minor M_ij multiplied by the checkerboard sign (-1)^(i + j).
• adj(A): the classical adjoint, equal to C^T.
• det(A): the determinant of A. The inverse A^-1 is defined only when det(A) is not zero.

For a 2x2 matrix A = [[a, b], [c, d]] the adjugate is the swap-and-sign pattern [[d, -b], [-c, a]]. For a 3x3 matrix the calculator evaluates nine 2x2 minors, multiplies each by the matching sign, lays them out as the cofactor matrix C, then transposes C to get adj(A) = C^T.

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.

For the 3x3 determinant that the cofactor expansion uses in this page, the cross product calculator is a fast way to see the same scalar from a vector input.

Four ideas cover the entire cofactor-to-inverse pipeline the calculator runs. They are the same concepts that show up in any linear-algebra textbook, so a quick look at them makes the result grid feel like a worked solution rather than a black box.

These four ideas are the entire toolbox the calculator uses, and the only bookkeeping is the checkerboard sign pattern on the cofactors.

When the same 3x3 matrix shows up in a vector identity, the dot product calculator is the cleanest way to confirm the scalar you are dividing the adjugate by in the adjoint matrix calculator.

Pick the matrix size, type the entries, and read the adjugate, cofactor, and inverse in the right-hand panel. The numbers update on every keystroke, so you do not need a separate Solve button to see the result.

1. 1 Choose the matrix size: Select 2x2 to enter four entries, or 3x3 to enter nine entries.
2. 2 Type the matrix entries: Real numbers, decimals allowed. The calculation updates as you type.
3. 3 Read the cofactor matrix: The first result grid. Each entry is the matching 2x2 minor with the checkerboard sign.
4. 4 Read the adjugate: The transpose of the cofactor matrix, equal to C^T.
5. 5 Read the determinant and inverse: The determinant scalar is shown next to the adjugate; the inverse is the adjugate divided by that scalar.
6. 6 Reset or change the size: Reset restores the default 2x2 matrix; the size selector jumps straight to 3x3 without reloading.

Suppose a 3x3 matrix comes from a small structural Jacobian: A = [[2, 1, 0], [1, 3, 1], [0, 1, 2]]. Drop the entries into the 3x3 grid. The cofactor matrix is [[5, -2, 1], [-2, 4, -2], [1, -2, 5]], the adjugate is its transpose, the determinant is 8, and the inverse is adj(A) / 8.

When the same small system is better solved by row reduction than by adjugate, the elimination method calculator is the row-reduction counterpart to this adjoint matrix calculator.

The adjugate is one of the few formulas in linear algebra that connects every important scalar and matrix around A in a single line. The calculator keeps that whole picture on screen at once, so the user sees the determinant, the cofactors, the adjugate, and the inverse together.

• • 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 4-entry 2x2 input and the 9-entry 3x3 input, covering the two sizes that show up in homework, statics problems, and computer-graphics transforms.
• • Inverse through the adjugate identity: The inverse grid is computed as adj(A) / det(A), the textbook cofactor-based inverse and the alternative to Gaussian elimination for small matrices.
• • Singular-matrix warning built in: When the determinant is zero the inverse panel switches to a warning instead of producing division-by-zero artifacts.
• • Decimal-friendly arithmetic: Entries can be decimals like 1.5 or 0.25, and the result grid keeps four decimal places of precision so the cofactor pattern stays readable.

The result panel is laid out so the same identity A * adj(A) = det(A) * I is visible: the determinant in one row, the adjugate grid in the middle, and the inverse grid on the bottom. That visual structure is the fastest way to learn the formula.

If the inverse is then used to back-substitute the right-hand side of a linear system, the substitution method calculator is a useful check on the values this page produces.

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

• • The calculator is restricted to 2x2 and 3x3 square matrices. For 4x4 and larger the cofactor expansion is the same in principle, but the run time and the chance of a sign error make a row-reduction approach more practical.
• • The output is rounded for display, so a cofactor expansion that produces a clean fraction like 1/3 will show as 0.3333. Keep full precision through the calculation and round only the final value.
• • Numerical round-off in the determinant can hide a true singular matrix when det(A) is just above or below zero. The calculator treats only an exact zero as singular.

According to Wolfram MathWorld, the adjugate of a matrix A satisfies A * adj(A) = det(A) * I, and for an invertible matrix the inverse is adj(A) divided by det(A).

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 2x2 or 3x3 inverse feeds the normal equations in a small regression, the linear regression calculator is the next page worth opening after this adjoint matrix calculator.