Adjoint Matrix Solver

Math Linear Algebra • Determinants and Rank

Written by STEM Calculators Team Published February 22, 2026 Updated February 24, 2026

Compute the adjoint (classical adjugate) \(\operatorname{adj}(A)\), defined as the transpose of the cofactor matrix: \(\operatorname{adj}(A)=C^\mathsf{T}\). If \(\det(A)\neq 0\), the tool also returns the inverse via \(A^{-1}=\operatorname{adj}(A)/\det(A)\).

Size (n×n)

Display decimals

Zero tolerance

Steps

Spotlight row i

Spotlight col j

Tip: click a matrix cell to set the spotlight \((i,j)\). Entries accept -3.5, 2e-4, fractions like 7/3, and pi, e.

Matrix \(A\)

2×2

Spotlight cell \((i,j)\) is red; its deleted row/column are gray; the remaining entries form the minor submatrix.

Results

\(\det(A)\)

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Singular?

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Spotlight \(C_{ij}\)

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Transpose placement

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Cofactor matrix \(C\)

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Adjoint \(\operatorname{adj}(A)=C^\mathsf{T}\)

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Inverse (bonus)

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Ready

Cofactor → transpose mapping

The spotlight cofactor \(C_{ij}\) lives in the cofactor matrix at \((i,j)\), then moves to \((j,i)\) in \(\operatorname{adj}(A)\).

Drag to pan • wheel to zoom • double-click to reset view

Enter a matrix and click “Calculate”.

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Theory

What “adjoint” means here (classical adjugate)

In linear algebra, the word adjoint can mean different things depending on context. This calculator computes the classical adjoint (also called the adjugate) of a square matrix \(A\), written \(\operatorname{adj}(A)\).

The classical adjoint is defined using cofactors: \[ \operatorname{adj}(A) = C^\mathsf{T}, \] where \(C\) is the cofactor matrix of \(A\). It is a purely algebraic construction and is closely tied to the determinant.

Minors and cofactors

For a matrix \(A\in\mathbb{R}^{n\times n}\) and an entry \(a_{ij}\), define the \((n-1)\times(n-1)\) submatrix \(B\) by deleting row \(i\) and column \(j\). The minor is the determinant of that submatrix: \[ M_{ij}=\det(B). \]

The cofactor applies a checkerboard sign: \[ C_{ij}=(-1)^{i+j} M_{ij}. \] The sign alternates by row and column: \[ \begin{bmatrix} + & - & + & - & \cdots\\ - & + & - & + & \cdots\\ + & - & + & - & \cdots\\ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix}. \]

Cofactor matrix and the transpose step

The cofactor matrix is \(C=[C_{ij}]\). The classical adjoint is its transpose: \[ \operatorname{adj}(A)=C^\mathsf{T}. \] Concretely, the entry in row \(r\), column \(c\) of \(\operatorname{adj}(A)\) is \[ \operatorname{adj}(A)_{rc}=C_{cr}. \]

This transpose is not a cosmetic step: it is what makes the key identity \(A\,\operatorname{adj}(A)=\det(A)I\) work correctly. The calculator’s canvas highlights how a chosen cofactor \(C_{ij}\) moves to \((j,i)\) in \(\operatorname{adj}(A)\).

Why adj(A) gives the inverse

A fundamental identity is: \[ A\,\operatorname{adj}(A)=\operatorname{adj}(A)\,A=\det(A)\,I. \] When \(\det(A)\neq 0\), you can divide both sides by \(\det(A)\) to obtain: \[ A^{-1}=\frac{\operatorname{adj}(A)}{\det(A)}. \]

This is sometimes called the classical inverse formula. It is excellent for theory and small matrices, and it explains why singular matrices (\(\det(A)=0\)) cannot have an inverse.

Special case: 2×2 adjoint

For \[ A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}, \] the cofactor matrix is \[ C=\begin{bmatrix} d & -c\\ -b & a\end{bmatrix}, \] and transposing gives \[ \operatorname{adj}(A)=\begin{bmatrix} d & -b\\ -c & a\end{bmatrix}. \]

This matches the familiar inverse formula \[ A^{-1}=\frac{1}{ad-bc}\begin{bmatrix} d & -b\\ -c & a\end{bmatrix}, \] whenever \(ad-bc\neq 0\).

Computational notes and numerical rank intuition

Computing \(\operatorname{adj}(A)\) directly requires many determinants (one for each minor). For a general \(n\times n\) matrix, that can be expensive as \(n\) grows, which is why this tool caps \(n\) at a moderate size.

With floating-point entries, very small values can occur due to measurement noise or rounding. The calculator uses a tolerance to treat \(|v|\le \varepsilon\) as zero in elimination steps, which can prevent misleading “almost-zero” determinants from producing unstable inverses.

If your determinant is very close to zero, the matrix may be numerically singular: the exact inverse (in theory) is extremely sensitive, and tiny perturbations in data can drastically change the result.

University note: pseudoinverse and why adj(A) is not used there

For non-square matrices or rank-deficient problems, the relevant object is often the Moore–Penrose pseudoinverse, not \(\operatorname{adj}(A)\). The pseudoinverse is defined through orthogonality and least-squares conditions (typically computed using SVD), and it behaves well when data is noisy.

The classical adjoint is still valuable: it explains determinant-based identities, cofactor expansions, and inverse formulas, and it connects to geometric scaling via \(\det(A)\).

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