Adjoint of a Matrix – Explained

The adjoint of a matrix is a fundamental operation in linear algebra.

Given a square matrix, the adjoint (also known as the adjugate or classical adjoint) produces a new matrix by taking the transpose of the matrix of cofactors.

In linear algebra, the adjoint of a matrix is a concept used to find the inverse of a square matrix.

The process involves taking the transpose of the matrix of cofactors.

 

Let’s delve deeper into this concept.

Consider a square matrix $A$ of size $n \times n$.

The cofactor of element $a_{ij}$, denoted as $C_{ij}$, is the determinant of the matrix obtained by removing the $i$th row and $j$th column from $A$.

The adjoint of the matrix $A$, denoted as $\text{adj}(A)$ or $A^*$, is obtained by taking the transpose of the matrix of cofactors:

\[
\text{adj}(A) =
\begin{bmatrix}
C_{11} & C_{21} & \cdots & C_{n1} \\
C_{12} & C_{22} & \cdots & C_{n2} \\
\vdots & \vdots & \ddots & \vdots \\
C_{1n} & C_{2n} & \cdots & C_{nn} \\
\end{bmatrix}^T
\]

 

Example

Let’s calculate the adjoint of the following $2 \times 2$ matrix $A$:

\[
A =
\begin{bmatrix}
3 & 1 \\
-2 & 4 \\
\end{bmatrix}
\]

First, find the cofactors:
\[
C_{11} = \begin{vmatrix} 4 \end{vmatrix} = 4, \quad C_{12} = \begin{vmatrix} -2 \end{vmatrix} = -2, \quad C_{21} = \begin{vmatrix} 1 \end{vmatrix} = 1, \quad C_{22} = \begin{vmatrix} 3 \end{vmatrix} = 3
\]

Next, create the matrix of cofactors:
\[
\text{adj}(A) =
\begin{bmatrix}
4 & 1 \\
-2 & 3 \\
\end{bmatrix}
\]

Finally, take the transpose to find the adjoint:
\[
\text{adj}(A) =
\begin{bmatrix}
4 & -2 \\
1 & 3 \\
\end{bmatrix}
\]

 

Summary of Steps

The adjoint of a matrix plays a crucial role in finding the inverse of a square matrix.

It is computed by taking the transpose of the matrix of cofactors, which are obtained by finding the determinants of smaller matrices formed by removing individual rows and columns.