### $2 \times 2$ Square Matrix

Let $\mathbf A$ be the square matrix of order $2$:

$\mathbf A = \begin {pmatrix} a & b \\ c & d \end {pmatrix}$

Then the adjugate matrix of $\mathbf A$ is:

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

### $3 \times 3$ Square Matrix

Let $\mathbf A$ be the square matrix of order $3$:

$\mathbf A = \begin {pmatrix} a_{1 1} & a_{1 2} & a_{1 3} \\ a_{2 1} & a_{2 2} & a_{2 3} \\ a_{3 1} & a_{3 2} & a_{3 3} \end {pmatrix}$

Let $A_{i j}$ denote the cofactor of element $a_{ij}$.

Then the adjugate matrix of $\mathbf A$ is:

$\adj {\mathbf A} = \begin {pmatrix} A_{1 1} & A_{2 1} & A_{3 1} \\ A_{1 2} & A_{2 2} & A_{3 2} \\ A_{1 3} & A_{2 3} & A_{3 3} \end {pmatrix}$

### Arbitrary Matrix $1$

Let $\mathbf A$ be the square matrix:

$\mathbf A = \begin {pmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end {pmatrix}$

Then the adjugate matrix of $\mathbf A$ is:

$\adj {\mathbf A} = \begin {pmatrix} 1 & -1 & 1 & -1 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 1 \\ \end {pmatrix}$

### Arbitrary Matrix $2$

Let $\mathbf A$ be the square matrix:

$\mathbf A = \begin {pmatrix} 1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & -1 & 0 \\ \end {pmatrix}$

Then the adjugate matrix of $\mathbf A$ is:

$\adj {\mathbf A} = \begin {pmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ \end {pmatrix}$

### Arbitrary Matrix $3$

Let $\mathbf A$ be the square matrix:

$\mathbf A = \begin {pmatrix} 1 & 2 & 0 \\ 0 & -1 & 2 \\ -1 & 2 & 0 \end {pmatrix}$

Then the adjugate matrix of $\mathbf A$ is:

$\adj {\mathbf A} = \begin {pmatrix} -4 & 0 & 4 \\ -2 & 0 & -2 \\ -1 & -4 & -1 \end {pmatrix}$

### Arbitrary Matrix $4$

Let $\mathbf A$ be the square matrix:

$\mathbf A = \begin {pmatrix} -1 & 2 & 0 \\ 0 & 1 & 3 \\ 2 & -3 & 3 \end {pmatrix}$

Then the adjugate matrix of $\mathbf A$ is:

$\adj {\mathbf A} = \begin {pmatrix} 12 & 6 & -2 \\ -6 & -3 & 1 \\ 6 & 3 & -1 \end {pmatrix}$