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Adjugate Matrix Calculator

Compute adj(A) via minors, cofactors, and transpose for 2x2 and 3x3 matrices

Enter 2x2 Matrix

Adjugate Formula

adj(A) = C^T (cofactor matrix transpose)

2x2: adj([[a,b],[c,d]]) = [[d,-b],[-c,a]]

C_ij = (-1)^(i+j) x M_ij

A^{-1} = adj(A) / det(A)

The adjugate (classical adjoint) of a square matrix is the transpose of its cofactor matrix. It is used to compute the matrix inverse and is important in linear algebra for solving systems of equations and understanding matrix properties.

⚠ If det(A) = 0, the matrix is singular. The adjugate still exists but cannot be used to compute an inverse. adj(A) will be rank deficient.

What is an Adjugate Matrix?

The adjugate (or classical adjoint) of a square matrix is the transpose of its cofactor matrix. It plays a crucial role in matrix theory because A x adj(A) = adj(A) x A = det(A) x I. This property allows computing the inverse as A^{-1} = adj(A)/det(A). For 2x2 matrices, the pattern is simple: swap diagonal entries, negate off-diagonals.

Minor

The minor M_ij is the determinant of the submatrix after removing row i and column j. For 2x2, each elements minor is the opposite diagonal element.

Cofactor

C_ij = (-1)^(i+j) x M_ij. The checkerboard sign pattern: + - + / - + - / + - + for 3x3.

Transpose

adj(A) = C^T. Swap rows and columns of the cofactor matrix. Row i column j of adj = C_ji.

2x2 Shortcut

adj([[a,b],[c,d]]) = [[d,-b],[-c,a]]. No minors needed for 2x2. Swap main diagonal, negate off-diagonals.

Teaching Example: 2x2 matrix A = [[1,2],[3,4]].
1. Minor matrix M: [[4,3],[2,1]]. Cofactors C: +4, -3, -2, +1
2. Transpose: adj(A) = [[4,-2],[-3,1]].
3. det(A) = 1*4 - 2*3 = -2. Verify: A x adj(A) = det(A) x I = -2I.

Applications

Matrix Inverse Linear Systems Cramers Rule Eigenvalue Calc Computer Graphics

Frequently Asked Questions

How to find adjugate?▼

1. Minors (2x2 dets). 2. Cofactors (checkerboard signs). 3. Transpose = adj(A).

Adjugate vs inverse?▼

A^{-1} = adj(A)/det(A). The adjugate divided by determinant gives the inverse.

2x2 adjugate pattern?▼

adj([[a,b],[c,d]]) = [[d,-b],[-c,a]]. Swap main, negate off-diagonal.

Singular matrix adjugate?▼

Exists but cannot compute inverse. adj(A) is rank deficient when det(A)=0.

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