Adjoint (Adjugate) Matrix Calculator
The Adjoint Matrix Calculator finds the adjugate (classical adjoint) of a 2×2 matrix. For a matrix [[a, b], [c, d]], the adjugate is the transpose of the cofactor matrix, which equals [[d, −b], [−c, a]]. The calculator also reports the determinant, since the inverse equals the adjugate divided by the determinant.
Formula
adj([[a, b], [c, d]]) = [[d, −b], [−c, a]]
- The adjugate of a 2×2 matrix swaps the two diagonal entries (a and d) and negates the two off-diagonal entries (b and c).
- It is the transpose of the cofactor matrix, not the inverse.
- The determinant det = ad − bc is shown because the inverse equals adjugate ÷ determinant when det ≠ 0.
- If the determinant is zero, the matrix is singular and has no inverse, though the adjugate still exists.
Example Calculation
Inputs
- a (row 1, col 1): 1
- b (row 1, col 2): 2
- c (row 2, col 1): 3
- d (row 2, col 2): 4
For [[1, 2], [3, 4]], the adjugate is [[4, −2], [−3, 1]] and the determinant is (1×4 − 2×3) = −2, so the inverse is the adjugate divided by −2.
Frequently asked questions
What is the adjugate of a matrix?
The adjugate (classical adjoint) is the transpose of the cofactor matrix. For a 2×2 matrix [[a,b],[c,d]] it is [[d,−b],[−c,a]].
Is the adjugate the same as the inverse?
No. The inverse equals the adjugate divided by the determinant. The adjugate itself involves no division, so it exists even for singular matrices.
How do I get the inverse from this?
Divide every entry of the adjugate by the determinant. If the determinant is zero, no inverse exists.
What is the cofactor matrix?
It replaces each entry with its signed minor. Transposing it gives the adjugate; for a 2×2 this reduces to swapping a/d and negating b/c.
Does this work for larger matrices?
This calculator covers 2×2 matrices. Larger matrices use the same cofactor-and-transpose definition but require more terms.