Adjugate Calculator (2x2)

This adjugate calculator finds the adjugate, also known as the classical adjoint, of a 2 by 2 matrix, a step at the heart of finding a matrix inverse by hand. The adjugate matters in linear algebra, engineering and computer science whenever you need to invert a small matrix, because it is built from cofactors alone and needs only one division at the end to give the full inverse. You enter the four entries of your matrix, labelled a, b, c and d, representing the top row (a, b) and the bottom row (c, d). The calculator returns the adjugate matrix, formed by swapping the two diagonal entries and negating the two off-diagonal entries, plus the determinant of your original matrix and a clear yes or no on whether an inverse exists, since a zero determinant means the matrix is singular. Results update live as you type, so you can check working done by hand or see how changing one entry changes the adjugate and determinant together. This suits students checking homework, anyone revising matrix algebra before an engineering, statistics or computer science course, and quick verification when solving simultaneous equations or transforming coordinates, where the inverse of a 2 by 2 matrix is often needed. Below you will find the formula, a worked example matching the default values, and answers to common questions about how adjugate and adjoint relate.

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adj = [[4, -2], [-3, 1]]

Determinant-2

Inverse exists?Yes

The formula

For the 2 by 2 matrix with rows (a, b) and (c, d), the adjugate is the matrix with rows (d, minus b) and (minus c, a). It satisfies A times adj(A) = det(A) times the identity, so the inverse is adj(A) divided by det(A) when the determinant is non-zero.

Worked example

For rows (1, 2) and (3, 4), the adjugate has rows (4, minus 2) and (minus 3, 1), and the determinant is minus 2, so the inverse is that adjugate divided by minus 2. Enter 1, 2, 3, 4 to confirm.

Frequently asked questions

What is the adjugate?

The transpose of the cofactor matrix. For a 2 by 2 it swaps the diagonal entries and negates the off-diagonal ones.

How does it give the inverse?

The inverse is the adjugate divided by the determinant, valid when the determinant is non-zero.

Is adjoint the same as adjugate?

In this classical sense yes, though adjoint also has a different meaning for conjugate transposes in other contexts.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.