What this calculator does
The adjugate of a 2×2 matrix, sometimes still called the adjoint, is formed by swapping the two entries on the main diagonal and flipping the sign of the two off-diagonal entries. For A = [[a, b], [c, d]], the adjugate is [[d, −b], [−c, a]]. It is a fixed, mechanical rearrangement, not a calculation that depends on anything else about the matrix.
Most people reach for the adjugate as a stepping stone to the inverse, since A⁻¹ = adj(A) / det(A). This calculator shows the adjugate itself as the main result, for the common case of a textbook question that asks for the adjugate specifically, rather than the full inverse a step later.
The formula
Take the matrix A = [[a, b], [c, d]]. Swap a and d to form the new diagonal, then negate b and c to form the new off-diagonal entries. The result, [[d, −b], [−c, a]], is the adjugate. Dividing every entry of the adjugate by the determinant, ad − bc, gives the inverse of A, provided the determinant is not zero.
| Term | Meaning |
|---|---|
| A | The original 2×2 matrix, entered as a, b, c, d reading left to right, top to bottom. |
| adj(A) | The adjugate matrix, [[d, −b], [−c, a]]. |
| det(A) | The determinant of A, equal to ad − bc. |
The inputs explained
| Field | What to enter |
|---|---|
| Row 1, Col 1 (a) | Top-left entry of the matrix. |
| Row 1, Col 2 (b) | Top-right entry of the matrix. |
| Row 2, Col 1 (c) | Bottom-left entry of the matrix. |
| Row 2, Col 2 (d) | Bottom-right entry of the matrix. |
When to use it
A textbook question asking for the adjugate directly
Some coursework asks for the adjoint or adjugate as the final answer, without going on to divide by the determinant, so having it labelled as its own result avoids doing that division and undoing it again.
Checking a hand-worked inverse
Since the inverse is just the adjugate divided by the determinant, seeing both numbers separately makes it easy to spot whether an error crept in during the division step or in forming the adjugate itself.
A singular matrix
When the determinant is zero, the matrix has no inverse, but the adjugate still exists and is still a valid, well-defined matrix in its own right.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the adjugate changes as the top-left entry (a) varies
Only the top-left entry a changes; note that a affects the determinant but not the adjugate's own top-left entry, since adj(A) puts d there instead.
| a | Adjugate matrix | Determinant of A |
|---|---|---|
| 1 | [ 6.000, -7.000 ; -2.000, 1.000 ] | -8.000 |
| 2 | [ 6.000, -7.000 ; -2.000, 2.000 ] | -2.000 |
| 4 | [ 6.000, -7.000 ; -2.000, 4.000 ] | 10.000 |
| 6 | [ 6.000, -7.000 ; -2.000, 6.000 ] | 22.000 |
| 8 | [ 6.000, -7.000 ; -2.000, 8.000 ] | 34.000 |
| 10 | [ 6.000, -7.000 ; -2.000, 10.000 ] | 46.000 |
How the adjugate changes as the top-right entry (b) varies
Only the top-right entry b changes.
| b | Adjugate matrix | Determinant of A |
|---|---|---|
| 1 | [ 6.000, -1.000 ; -2.000, 4.000 ] | 22.000 |
| 3 | [ 6.000, -3.000 ; -2.000, 4.000 ] | 18.000 |
| 5 | [ 6.000, -5.000 ; -2.000, 4.000 ] | 14.000 |
| 7 | [ 6.000, -7.000 ; -2.000, 4.000 ] | 10.000 |
| 9 | [ 6.000, -9.000 ; -2.000, 4.000 ] | 6.000 |
| 11 | [ 6.000, -11.000 ; -2.000, 4.000 ] | 2.000 |
Questions
Is the adjugate the same as the transpose?
No. For a 2×2 matrix the adjugate happens to look similar to a transpose with sign changes, but the two operations are different in general and only coincide in this specific pattern for the 2×2 case.
What is the adjugate used for beyond finding the inverse?
It appears in Cramer's rule for solving small systems of linear equations, and in some formulas for cofactor expansion in larger matrices, where it generalises to a matrix of signed minors.
Does the adjugate exist for a matrix with determinant zero?
Yes. The adjugate is defined regardless of the determinant. It is only the inverse, adjugate divided by determinant, that fails to exist when the determinant is zero.
Is "adjoint" the same word as "adjugate"?
In the context of a real matrix like this one, yes, older textbooks use "adjoint" for exactly what modern texts call the "adjugate". A different, unrelated meaning of "adjoint" exists for complex matrices and linear operators, referring to the conjugate transpose.
To take this the extra step and get the full inverse matrix, see the 2x2 matrix inverse calculator. For other matrix arithmetic, the 2x2 matrix operations calculator covers addition, subtraction and multiplication.