Enter the elements of a 2×2 or 3×3 square matrix and the Adjoint Matrix Calculator computes the adjugate (classical adjoint) — the transpose of the cofactor matrix. Fill in your matrix entries, select the matrix size, and get back the full adjoint matrix along with intermediate cofactor values.
Determinant of Matrix
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Adj[1,1]
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Adj[1,2]
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Adj[1,3]
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Adj[2,1]
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Adj[2,2]
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Adj[2,3]
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Adj[3,1]
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Adj[3,2]
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Adj[3,3]
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Matrix Invertible?
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The adjoint of a matrix, also called the adjugate, is the transpose of its cofactor matrix. For each element of the original matrix, you calculate the corresponding cofactor (signed minor), arrange them into the cofactor matrix, then transpose it. The result is the adjoint matrix.
The inverse of a matrix A is given by A⁻¹ = adj(A) / det(A). So the adjoint is closely related to the inverse — it is the inverse scaled by the determinant. If the determinant is zero, the matrix has no inverse, but the adjoint can still be computed.
For a 2×2 matrix [[a, b], [c, d]], the adjoint is simply [[d, -b], [-c, a]]. You swap the diagonal elements and negate the off-diagonal elements. No cofactor expansion is needed for 2×2 matrices.
For a 3×3 matrix, compute the cofactor of each element by taking the determinant of the 2×2 submatrix formed by deleting that element's row and column, then applying the sign pattern (+, -, +) / (-, +, -) / (+, -, +). Arrange these cofactors into a 3×3 cofactor matrix, then transpose it to get the adjoint.
The adjoint matrix is the zero matrix when every 2×2 minor of the original matrix is zero. This happens when the rank of the matrix is less than or equal to n−2, where n is the matrix dimension. For a 2×2 matrix, this means all entries would need to be zero.
A key identity is: A × adj(A) = det(A) × I, where I is the identity matrix. This means multiplying a matrix by its adjoint yields a scalar multiple of the identity matrix, with the scalar being the determinant. This property is used to derive the matrix inverse formula.
No. The adjoint (adjugate) is only defined for square matrices. Non-square matrices do not have cofactors in the classical sense, so the adjugate cannot be formed. For non-square matrices, related concepts like the Moore-Penrose pseudoinverse are used instead.
If det(A) = 0, the matrix is singular — it has no inverse. The adjoint can still be calculated, but you cannot use the formula A⁻¹ = adj(A) / det(A) since division by zero is undefined. A zero determinant indicates linearly dependent rows or columns.