Adjugate Matrix¶

Definition¶

The adjugate matrix $\operatorname{adj}(\boldsymbol{A})$ is the transpose of the cofactor matrix of a square matrix $\boldsymbol{A}$.

Interpretation¶

The adjugate collects all cofactors $C_{ij}$ of a square matrix in one matrix, transposed, so that multiplying it with $\boldsymbol{A}$ gives the determinant times the identity:

The same holds for $\operatorname{adj}(\boldsymbol{A})\boldsymbol{A}$, and dividing by $\det(\boldsymbol{A})\neq0$ gives the inverse formula in the Key Results.

The transpose is what makes this work. Entry $(i,j)$ of $\boldsymbol{A}\operatorname{adj}(\boldsymbol{A})$ is $\displaystyle \sum_k a_{ik}C_{jk}$. For $i=j$ this is the cofactor expansion of $\det(\boldsymbol{A})$ along row $i$. For $i\neq j$ it is the expansion of the determinant of the matrix obtained from $\boldsymbol{A}$ by replacing row $j$ with row $i$, which is $0$ because that matrix has two equal rows. For a $2\times2$ matrix with $ad-bc\neq0$ this gives $\boldsymbol{A}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$$\frac{1}{ad-bc}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$.

The formula is mainly of theoretical use: it shows that every entry of $\boldsymbol{A}^{-1}$ is a ratio of determinants, and applied to $\boldsymbol{x}=\boldsymbol{A}^{-1}\boldsymbol{b}$ it gives Cramer's rule. For larger matrices, computing all $n^2$ cofactors is far more work than Gauss-Jordan elimination on $(\boldsymbol{A}\mid\boldsymbol{I})$$\mathbf{I})$. A common slip is to forget the transpose or the signs $(-1)^{i+j}$: for a $2\times2$ matrix the untransposed cofactor matrix is $\begin{pmatrix}d&-c\\-b&a\end{pmatrix}$, not the adjugate.

Key Results¶

• If $\boldsymbol{A}$ is nonsingular, then $\boldsymbol{A}^{-1}=\frac{1}{\det(\boldsymbol{A})}\operatorname{adj}(\boldsymbol{A})$$\frac{1}{\det (\mathbf{A})}\mathrm{adj}(\mathbf{A})$.
• The adjugate is defined for every square matrix, even if the inverse does not exist.

Examples¶

• For $\boldsymbol{A}=\begin{pmatrix}a&b\\c&d\end{pmatrix}$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, $\operatorname{adj}(\boldsymbol{A})=\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$$\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$.
• If $\boldsymbol{A}=\boldsymbol{I}_n$${\mathbf{I}}_n$, then $\operatorname{adj}(\boldsymbol{A})=\boldsymbol{I}_n$${\mathbf{I}}_n$.

Prerequisites¶

• Cofactor Expansion
• Transpose

Related Concepts¶

• Determinant
• Inverse and Singular Matrices
• Cramer's Rule
• Matrix Product

Sources¶

• Math II: Linear Algebra · Week 4

Aliases: adjugate, classical adjoint Keywords: adjoint matrix Symbols: sum

Last updated: 20 October 2026 03:02