Adjugate Matrix
Definition
Let \(A\) be a square matrix of size \(n\times n\). The adjugate matrix of \(A\), denoted by \( \operatorname{adj}(A) \), is the transpose of the matrix of cofactors of \(A\). That is, we first compute the matrix of cofactors and then take its transpose.
Minors and Cofactors
Minor \(M_{ij}\)
The minor \(M_{ij}\) is the determinant of the submatrix obtained by deleting row \(i\) and column \(j\) from the matrix \(A\).
Cofactor \(C_{ij}\)
The cofactor is defined as
$$C_{ij} = (-1)^{i+j}M_{ij}$$
The factor \( (-1)^{i+j} \) produces the alternating pattern of signs used in the cofactor matrix.
Construction of the Adjugate Matrix
If \(C\) is the matrix of cofactors
$$ C = \begin{pmatrix} C_{11} & C_{12} & \cdots \\ C_{21} & C_{22} & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix} $$
then the adjugate matrix is
$$\operatorname{adj}(A)=C^{T} = \begin{pmatrix} C_{11} & C_{21} & \cdots \\ C_{12} & C_{22} & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix} $$
Fundamental Property
A key identity involving the adjugate matrix is
$$A\,\operatorname{adj}(A)=(\det A)I$$
and also
$$\operatorname{adj}(A)\,A=(\det A)I$$
where \(I\) is the identity matrix.
Finding the inverse matrix
If \(\det(A)\neq 0\) then the matrix \(A\) is invertible and its inverse can be computed using
$$A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A)$$
This method is known as the adjugate method.
Example (\(2 \times 2\) Case)
If
$$ A= \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$
then
$$ \operatorname{adj}(A)= \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$
and therefore
$$ A^{-1}= \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$
Interactive Examples
In this applet you will find examples of how to compute the adjugate matrix. Press the “New” button to generate a different example, move the level slider to increase or decrease the difficulty of the exercise, and use the arrow buttons to move forward or backward through the steps of the procedure for computing the adjugate matrix.