Enter your matrix
Pick 2×2, 3×3, or 4×4. Coefficients use row letters (a₁, b₂, …) so each position stays clear on any screen.
Linear algebra made practical
Adjoint Matrix Calculator
Calculate, understand, and apply adjoint matrices for determinants, cofactors, and matrix inverses. Use the tool below, then follow the guides for formulas, examples, and common mistakes.
Get adj(A) instantly
The calculator forms cofactors, applies sign patterns, and transposes to produce the classical adjugate used with det(A) and A⁻¹.
Learn the method
Scroll the sections below for definitions, formulas, worked examples, and links to deeper articles on inverses and determinants.
Community rating 4.8/5 · 96 reviews
Adjoint Matrix Calculator
Enter integers, decimals, or fractions (example: 3/4). Fill every cell to update the adjoint matrix output.
Matrix input ()
Adjoint matrix ()
Fill all matrix entries to compute the adjoint.
Use valid numbers or fractions (example: -2, 0.5, 3/4).
adj(A) is the transpose of the cofactor matrix. When det(A) ≠ 0, you can form A⁻¹ = adj(A) / det(A).
How to use this calculator
- Select 2×2, 3×3, or 4×4.
- Type each value in the labeled input grid.
- Read the adjoint in the result grid labeled c₁₁, c₁₂, and so on.
- Use Clear entries to reset, or scroll down for formulas and examples.
What Is an Adjoint Matrix?
The adjoint matrix (also called the adjugate) of a square matrix A is built from cofactors. First form the cofactor matrix C, where each entry Cij combines a minor with a sign factor (−1)i+j. Then adj(A) = CT, the transpose of C.
In many linear algebra courses, adj(A) is the partner of the determinant. For an n×n matrix, A · adj(A) = det(A) · I. That identity is why adj(A) appears when you solve systems, invert matrices, or apply Cramer's rule.
Adjoint versus adjugate: the names refer to the same cofactor-transpose matrix in this real-matrix setting. Some advanced texts use "adjoint" for the conjugate transpose on complex matrices; on this site we mean the classical adjugate unless stated otherwise.
Real-world uses include solving small linear systems by hand, checking software output in control and signal courses, and bridging determinant ideas to invertibility in engineering mathematics. For a focused walkthrough, read what is an adjoint matrix.
Definition
adj(A) is the transpose of the matrix of cofactors of A.
Cofactor link
Each cofactor uses a minor (determinant of a submatrix) and a checkerboard sign.
Applications
Inverses, determinants, and solvability checks for square systems.
Adjoint Matrix Formula
Cofactor: C_ij = (−1)^(i+j) · M_ij
Cofactor matrix: C = [C_ij]
Adjoint: adj(A) = C^T
Determinant link (n×n): A · adj(A) = det(A) · I
2×2 shortcut for A = [[a, b], [c, d]]:
adj(A) = [[d, −b], [−c, a]]
Matrix notation often writes adj(A) instead of adjugate(A). The transpose step is easy to forget: you compute cofactors in place, then swap rows and columns.
The formula adj(A) = CT is the definition used by the calculator above. See the adjoint matrix formula article for notation and determinant relationships.
How to Find the Adjoint Matrix
Use this step-by-step method for any square size, then confirm with the calculator at the top of the page.
Confirm the matrix is square
Adjoint is defined for n×n matrices. Label rows and columns so minors stay organized.
Compute each minor M_ij
Delete row i and column j, then take the determinant of the remaining (n−1)×(n−1) matrix.
Build cofactors with signs
C_ij = (−1)^(i+j) · M_ij. The sign pattern alternates like a checkerboard starting with + at (1,1).
Assemble the cofactor matrix C
Place all C_ij in the same positions as the original entries.
Transpose to get adj(A)
Swap rows and columns: adj(A) = C^T. Compare with the calculator output.
Adjoint Matrix Examples
Try these in the calculator, then read the full <a href="/blog/adjoint-matrix-examples/" class="text-link underline">adjoint matrix examples</a> guide.
2×2 example
Let A = [[1, 2], [3, 4]].
adj(A) = [[4, −2], [−3, 1]], det(A) = −2
Adjoint: [[4, −2], [−3, 1]]
3×3 diagonal matrix
A = [[2, 0, 0], [0, 3, 0], [0, 0, 5]].
Zeros simplify many cofactors.
Adjoint: adj(A) = [[15, 0, 0], [0, 10, 0], [0, 0, 6]]
Inverse check
When det(A) ≠ 0, A⁻¹ = (1/det(A)) · adj(A).
Compute adj(A) first, then divide every entry by det(A).
Adjoint: Use the calculator, then verify with the inverse section below.
Cofactor Matrix Calculator
Cofactors are the building blocks of the adjoint. The on-page tool computes adj(A), which is the transpose of the cofactor matrix. Understanding cofactors helps you interpret each sign and minor behind the result.
A minor M_ij is the determinant of the submatrix you get after removing row i and column j. The cofactor multiplies that minor by (−1)^(i+j). For practice with sign rules and minors, see cofactor matrix calculator.
Minor
Determinant of the matrix after deleting one row and one column.
Sign matrix
Alternating + and − by position: (+) at (1,1), (−) at (1,2), and so on.
Simplification
Zeros in A often reduce the number of nonzero cofactors you must compute.
Adjoint Matrix and Inverse Matrix
When A is invertible (det(A) ≠ 0), the adjoint gives a direct path to the inverse. Divide every entry of adj(A) by det(A).
If det(A) = 0, A is singular and cannot be inverted, even though you can still compute adj(A). Always check the determinant before applying the inverse formula.
A^−1 = (1 / det(A)) · adj(A), when det(A) ≠ 0
Non-singular matrices
det(A) ≠ 0 guarantees A⁻¹ exists and the formula above applies.
Singular matrices
det(A) = 0: no inverse, but adj(A) may still be useful in theory exercises.
Verification
Multiply A · A⁻¹; you should get the identity matrix within rounding tolerance.
Adjoint Matrix vs Transpose Matrix
Students often mix up AT and adj(A) because both involve changing positions of entries. They answer different questions.
The transpose swaps rows and columns of A itself. The adjoint uses cofactors first, then transposes that cofactor matrix. They coincide only in special cases, not for a generic matrix.
| Topic | Transpose AT | Adjoint adj(A) |
|---|---|---|
| Starting matrix | Uses entries of A directly | Uses cofactors of A |
| Operation | Swap row and column indices | Minors, signs, then transpose |
| Typical use | Symmetry, inner products, notation | Determinants, inverses, Cramer |
Determinants and Adjoint Matrices
The determinant measures whether A is invertible and scales volume under the linear map defined by A. The adjoint encodes complementary algebraic information through cofactor expansion.
Cofactor expansion along a row expresses det(A) as a sum involving entries of A and cofactors. That is why cofactor practice strengthens both determinant and adjoint skills.
Read more in determinants and adjoint matrices.
Determinant basics
det(A) is a scalar; for 2×2, ad − bc.
Cofactor expansion
Expand det(A) along a row or column using cofactors.
Solvability
det(A) ≠ 0 means unique solutions for Ax = b (for square A).
Adjoint Matrix Calculator
The interactive tool at the top of this page is the fastest way to obtain adj(A) for 2×2, 3×3, and 4×4 matrices. Enter coefficients, read the labeled adjoint grid, and clear entries to try new examples.
The calculator runs in your browser: no uploads, no account. Use it for homework checks, exam review, and side-by-side comparison with manual cofactor work.
For a dedicated article on features and study tips, visit adjoint matrix calculator guide.
Common Adjoint Matrix Mistakes
Avoid these errors when computing by hand or checking the tool.
Forgetting the final transpose
Stopping at the cofactor matrix C instead of C^T gives the wrong matrix.
Sign errors on cofactors
Track (−1)^(i+j) carefully; one wrong sign changes the whole row.
Confusing adj(A) with A^T
Transpose and adjoint are different unless you are in a special case.
Dividing by det(A) when det(A) = 0
You cannot form an inverse; check determinant first.
Deleting the wrong row or column for minors
Minor M_ij removes row i and column j of the original matrix.
Matrix Inverse Using Adjoint Method
The adjoint method is a standard textbook approach for small matrices. Compute det(A) and adj(A), then combine them when det(A) ≠ 0.
For larger systems, courses move to row reduction or numerical methods, but the adjoint method remains essential for understanding why inverses exist.
- Step 1: Compute det(A) If det(A) = 0, stop; A is not invertible.
- Step 2: Compute adj(A) Use cofactors and transpose, or the calculator above.
- Step 3: Scale by 1/det(A) Each entry of adj(A) is divided by det(A) to obtain A⁻¹.
- Step 4: Verify Multiply A · A⁻¹ and confirm you get I.
FAQs About Adjoint Matrices
What is the difference between adjoint and adjugate?
For real square matrices in introductory linear algebra, they name the same matrix: the transpose of the cofactor matrix.
How do I calculate the adjoint of a 2×2 matrix?
Swap the main diagonal and negate the off-diagonal: adj([[a,b],[c,d]]) = [[d,−b],[−c,a]].
Can I find the inverse from the adjoint?
Yes, when det(A) ≠ 0. Use A⁻¹ = adj(A) / det(A).
Is the adjoint the same as the transpose?
No. The transpose reorders entries of A. The adjoint uses cofactors, then transposes that matrix.
What sizes does the calculator support?
2×2, 3×3, and 4×4 square matrices.
Does the site store my matrix entries?
No. Calculations run locally in your browser.
Why is det(A) important for adjoint work?
det(A) tells you if A is invertible and appears in A · adj(A) = det(A) · I.
Where can I learn the full step-by-step method?
See How to Find the Adjoint Matrix on this page or the blog series starting with the formula guide.