Adjoint Matrix Calculator

Enter a 2×2 matrix to compute its determinant, adjoint (adjugate) matrix, and inverse using adj(A) = [[d, −b], [−c, a]] for A = [[a, b], [c, d]].

Quick Facts

Formula
adj(A) = transpose of the cofactor matrix
For A = [[a,b],[c,d]], adj(A) = [[d,-b],[-c,a]] — swap the diagonal, negate the off-diagonal.
Inverse link
A⁻¹ = adj(A) ÷ det(A)
Only valid when det(A) ≠ 0; a zero determinant means A is singular.

Your Results

Calculated
Determinant det(A)
-
a·d − b·c
Adjoint matrix adj(A)
-
Transpose of the cofactor matrix
Inverse matrix A⁻¹
-
adj(A) ÷ det(A)
Classification
-
Invertible or singular

Ready

Enter a 2×2 matrix and press Calculate.

How the Adjoint Matrix Calculator works

This tool computes the adjoint (also called the adjugate) of a 2×2 matrix, along with its determinant and inverse. For a square matrix A, the adjugate is the transpose of the matrix of cofactors, and it is the key ingredient in the classical formula for a matrix inverse.

Formula and method

For a 2×2 matrix A = [[a, b], [c, d]], the cofactors are C₁₁ = d, C₁₂ = −c, C₂₁ = −b, C₂₂ = a. Transposing the cofactor matrix gives the adjugate:

adj(A) = [[d, −b], [−c, a]]

In words: swap the two diagonal entries (a and d) and negate the two off-diagonal entries (b and c). The determinant is det(A) = a·d − b·c, and whenever det(A) ≠ 0 the inverse follows directly: A⁻¹ = adj(A) / det(A). This calculator computes all three — determinant, adjoint, and inverse — from the same four entries.

Common sources of error

  • Sign mistakes: the off-diagonal entries of adj(A) are negated — forgetting the minus sign on −b and −c is the most common error.
  • Confusing adjoint with transpose: the plain transpose of A is [[a, c], [b, d]]; the adjugate additionally applies the sign pattern of the cofactors, which for a 2×2 matrix also swaps the diagonal.
  • Dividing by a zero determinant: if det(A) = 0, A has no inverse — the adjoint matrix is still defined, but adj(A)/det(A) is undefined.

Checking your result

A reliable check is the identity A × adj(A) = det(A) × I, where I is the identity matrix. Multiply your original matrix by the computed adjoint; every off-diagonal entry of the product should be 0, and both diagonal entries should equal the determinant. If they don't match, recheck the sign of each cofactor.

Applications

The adjugate matrix appears in Cramer's rule for solving linear systems, in the closed-form inverse formula used here, and in symbolic or exact-fraction computations where dividing by the determinant too early would introduce rounding error. For matrices larger than 2×2, the same idea generalizes: each entry of adj(A) is a signed minor (cofactor) of A, transposed into position.

Frequently Asked Questions

What is the adjoint (adjugate) of a matrix?
The adjoint, or adjugate, of a square matrix A is the transpose of its cofactor matrix. For a 2×2 matrix [[a, b], [c, d]], the adjugate is [[d, −b], [−c, a]]: the diagonal entries are swapped and the off-diagonal entries are negated.
How is the adjoint related to the inverse?
The inverse of a matrix equals its adjoint divided by its determinant: A⁻¹ = adj(A) / det(A). This formula only works when det(A) is nonzero — if the determinant is zero, the matrix has no inverse even though its adjoint is still defined.
What does it mean if the determinant is zero?
A determinant of zero means the matrix is singular: it has no inverse, and its rows (or columns) are linearly dependent. The adjoint matrix can still be computed in this case, and it satisfies A × adj(A) equal to the zero matrix.
Is the adjoint the same as the transpose?
No. The transpose only swaps rows and columns (Aᵀ = [[a, c], [b, d]]). The adjugate additionally applies the sign pattern from the cofactor expansion, which for a 2×2 matrix means swapping the diagonal entries and negating the off-diagonal ones — a different matrix from the plain transpose except in special cases.

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