Adjoint Matrix
Calculate the adjoint (adjugate) of a 2×2, 3×3, or 4×4 matrix instantly. Enter your matrix values to get the exact result, free online.
How to Use the Matrix Adjoint Calculator
- Select the matrix size: 2×2, 3×3, or 4×4.
- Enter each value into the labeled matrix cells (a11, a12, a21, and so on).
- Click “Calculate.”
- View the resulting adjoint matrix instantly.
Features of the Matrix Adjoint Calculator
- Calculates the adjoint (adjugate) of any 2×2, 3×3, or 4×4 matrix.
- Uses cofactor expansion and matrix transposition for accurate results.
- Works with positive, negative, and decimal matrix entries.
- Free to use with no sign-up required.
Matrix Adjoint Formula
The adjoint (also called the adjugate) of a square matrix A is the transpose of its cofactor matrix.
Adjoint(A) = Transpose(Cofactor Matrix of A)
For a 2×2 matrix [[a, b], [c, d]], the adjoint simplifies to a direct formula: swap the diagonal entries and negate the off-diagonal entries.
Adjoint([[a, b], [c, d]]) = [[d, −b], [−c, a]]
2×2 Matrix Adjoint Reference Table
| Matrix A | Adjoint(A) |
|---|---|
| [[3, 4], [5, 6]] | [[6, −4], [−5, 3]] |
| [[2, 0], [0, 3]] | [[3, 0], [0, 2]] |
| [[1, 0], [0, 1]] | [[1, 0], [0, 1]] |
| [[4, 7], [2, 6]] | [[6, −7], [−2, 4]] |
How to Calculate the Adjoint of a 2×2 Matrix
For a 2×2 matrix, swap the two diagonal entries and change the sign of the two off-diagonal entries.
Suppose A = [[3, 4], [5, 6]]. Swap the diagonal entries 3 and 6, and negate the off-diagonal entries 4 and 5.
Adjoint(A) = [[6, −4], [−5, 3]].
Solved Examples on Matrix Adjoint
1. Adjoint of a 2×2 Matrix
Example: Find the adjoint of A = [[4, 7], [2, 6]].
Swap the diagonal (4 and 6) and negate the off-diagonal (7 and 2): Adjoint(A) = [[6, −7], [−2, 4]].
2. Adjoint of the Identity Matrix
Example: Find the adjoint of A = [[1, 0], [0, 1]].
Since the identity matrix is symmetric with a diagonal of 1s and zero off-diagonal entries, Adjoint(A) = [[1, 0], [0, 1]], the same as A.
3. Adjoint of a 3×3 Matrix
Example: Find the adjoint of A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]].
Computing the cofactor for each entry and transposing the result gives Adjoint(A) = [[−24, 18, 5], [20, −15, −4], [−5, 4, 1]].
4. Adjoint of a Diagonal Matrix
Example: Find the adjoint of A = [[2, 0], [0, 3]].
Swap the diagonal entries (2 and 3) and negate the off-diagonal entries, which are already zero: Adjoint(A) = [[3, 0], [0, 2]].
1. What is the adjoint of a matrix?
The adjoint (or adjugate) of a square matrix A is the transpose of its cofactor matrix. Each entry is calculated from the determinant of a smaller matrix formed by removing one row and one column, with alternating signs.
2. What’s the difference between the adjoint and the inverse of a matrix?
The inverse of a matrix equals its adjoint divided by its determinant: A⁻¹ = Adjoint(A) ÷ det(A). The adjoint is always defined, but the inverse only exists when the determinant is non-zero.
3. What is a cofactor matrix?
A cofactor matrix is formed by replacing each entry of a matrix with its cofactor: the determinant of the minor matrix obtained by deleting that entry’s row and column, multiplied by (−1) raised to the sum of the row and column indices.
4. Does every matrix have an adjoint?
Every square matrix has an adjoint, regardless of whether its determinant is zero. This is different from the inverse, which only exists when the determinant is non-zero.
5. Can you find the adjoint of a non-square matrix?
No. The adjoint is only defined for square matrices, since it relies on computing determinants of minors, which requires equal row and column counts.
6. Where is the matrix adjoint used?
The adjoint is used to compute matrix inverses, solve systems of linear equations via Cramer’s rule, and appears in computer graphics for transforming normal vectors.