Adjoint Matrix Calculator

What is an Adjoint (Adjugate) Matrix?

In linear algebra, the adjoint matrix (also known as the adjugate matrix or classical adjoint) of a square matrix $A$ is the transpose of its cofactor matrix. Denoted as $\text{adj}(A)$ or $\operatorname{adj}(A)$, it plays a pivotal role in matrix inversion, solving linear systems via Cramer's rule, and analytical derivations in multivariable calculus and physics.

The adjoint matrix is fundamentally connected to the determinant and inverse matrix through the identity:

$$A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = \det(A) \cdot I_n$$

When $\det(A) \neq 0$, the inverse of matrix $A$ is directly expressed as:

$$A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$$

For further matrix computations, explore our Matrix Inverse Calculator, Matrix Determinant Calculator, and Matrix Transpose Calculator.

Step-by-Step Method to Find the Adjoint Matrix

Calculating the adjoint of an $n \times n$ matrix involves three core steps:

1. Calculate the Matrix of Minors ($M$): For every element $a_{i,j}$ in matrix $A$, find its minor $M_{i,j}$, which is the determinant of the $(n-1) \times (n-1)$ submatrix created by deleting the $i$-th row and $j$-th column.
2. Form the Cofactor Matrix ($C$): Multiply each minor $M_{i,j}$ by the alternating sign factor $(-1)^{i+j}$: $$C_{i,j} = (-1)^{i+j} M_{i,j}$$
3. Transpose the Cofactor Matrix: Swap the rows and columns of $C$ to obtain $\text{adj}(A) = C^T$: $$[\text{adj}(A)]_{i,j} = C_{j,i}$$

Adjoint of a 2x2 Matrix Shortcut

For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the adjoint can be computed directly using a fast geometric rule:

1. Swap the main diagonal entries ($a$ and $d$).
2. Negate the off-diagonal entries (change $b$ to $-b$ and $c$ to $-c$).

$$\text{adj}\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

Worked Example (3x3 Matrix)

Consider the matrix:

$$A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix}$$

1. Matrix of Minors:

• $M_{1,1} = \det\begin{pmatrix} 1 & 4 \\ 6 & 0 \end{pmatrix} = (0 - 24) = -24$
• $M_{1,2} = \det\begin{pmatrix} 0 & 4 \\ 5 & 0 \end{pmatrix} = (0 - 20) = -20$
• $M_{1,3} = \det\begin{pmatrix} 0 & 1 \\ 5 & 6 \end{pmatrix} = (0 - 5) = -5$
• $M_{2,1} = \det\begin{pmatrix} 2 & 3 \\ 6 & 0 \end{pmatrix} = (0 - 18) = -18$
• $M_{2,2} = \det\begin{pmatrix} 1 & 3 \\ 5 & 0 \end{pmatrix} = (0 - 15) = -15$
• $M_{2,3} = \det\begin{pmatrix} 1 & 2 \\ 5 & 6 \end{pmatrix} = (6 - 10) = -4$
• $M_{3,1} = \det\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} = (8 - 3) = 5$
• $M_{3,2} = \det\begin{pmatrix} 1 & 3 \\ 0 & 4 \end{pmatrix} = (4 - 0) = 4$
• $M_{3,3} = \det\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = (1 - 0) = 1$

2. Cofactor Matrix:

$$C = \begin{pmatrix} +(-24) & -(-20) & +(-5) \\ -(-18) & +(-15) & -(-4) \\ +(5) & -(4) & +(1) \end{pmatrix} = \begin{pmatrix} -24 & 20 & -5 \\ 18 & -15 & 4 \\ 5 & -4 & 1 \end{pmatrix}$$

3. Adjoint Matrix $\text{adj}(A) = C^T$:

$$\text{adj}(A) = \begin{pmatrix} -24 & 18 & 5 \\ 20 & -15 & -4 \\ -5 & 4 & 1 \end{pmatrix}$$