# Adjoint of the Matrix (2×2 & 3×3) – Properties, Examples

Here you will learn how to find adjoint of the matrix 2×2 and 3×3, cofactors and its properties with examples.

Let’s begin –

Let A = $$[a_{ij}]$$ be a square matrix of order n and let $$C_{ij}$$ be a cofactor of $$a_{ij}$$ in A. Then the transpose of the matrix of cofactors of elements of A is called adjoint of A and is denoted by adj A.

Thus, adj A = $$[C{ij}]^T$$ $$\implies$$ $$(adj A)_{ij}$$ = $$C_{ij}$$ = Cofactor of $$a_{ij}$$ in A.

If A = $$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$ then,

adj A = $${\begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23}\\ C_{31} & C_{32} & C_{33} \end{bmatrix}}^T$$ = $$\begin{bmatrix} C_{11} & C_{21} & C_{31} \\ C_{12} & C_{22} & C_{32}\\ C_{13} & C_{23} & C_{33} \end{bmatrix}$$

where $$C_{ij}$$ denotes cofactor of $$a_{ij}$$ in A.

#### How to find Cofactors and Adjoint for 2×2 Matrix :

Example : Let A = $$[a_{ij}]$$ = $$\begin{bmatrix} p & q \\ r & s \end{bmatrix}$$

then, cofactor of $$a_{11}$$ = s

and cofactor of $$a_{12}$$ = -r

cofactor of $$a_{21}$$ = -q

cofactor of $$a_{22}$$ = p

$$\therefore$$  adj A = $${\begin{bmatrix} s & -r \\ -q & p \end{bmatrix}}^T$$ = $$\begin{bmatrix} s & -q \\ -r & p \end{bmatrix}$$

Rule : It is evident from this example that the adjoint of a square matrix of order 2 can be easily obtained by interchanging the diagonal elements and changing signs of off-diagonal elements.

If A = $$\begin{bmatrix} -2 & 3 \\ -5 & 4 \end{bmatrix}$$ then by the above rule, we obtain adj A $$\begin{bmatrix} 4 & -3 \\ 5 & -2 \end{bmatrix}$$

#### How to find Cofactors and Adjoint for 3×3 Matrix :

Example : Let A = $$[a_{ij}]$$ = $$\begin{bmatrix} 1 & 1 & 1 \\ 2 & 1 & -3 \\ -1 & 2 & 3 \end{bmatrix}$$

Let $$C_{ij}$$ be cofactor of $$a_{ij}$$ in A. Then the cofactors of elements of A are given by

$$C_{11}$$ = $$\begin{vmatrix} 1 & -3 \\ 2 & 3 \end{vmatrix}$$ = 9, $$C_{12}$$ = -$$\begin{vmatrix} 2 & -3 \\ -1 & 3 \end{vmatrix}$$ = -3,  $$C_{13}$$ = $$\begin{vmatrix} 2 & 1 \\ -1 & 2 \end{vmatrix}$$ = 5

$$C_{21}$$ = -$$\begin{vmatrix} 1 & 1 \\ 2 & 3 \end{vmatrix}$$ = -1, $$C_{22}$$ = $$\begin{vmatrix} 1 & 1 \\ -1 & 3 \end{vmatrix}$$ = 4,  $$C_{23}$$ = -$$\begin{vmatrix} 1 & 1 \\ -1 & 2 \end{vmatrix}$$ = -3

$$C_{31}$$ = $$\begin{vmatrix} 1 & 1 \\ 1 & -3 \end{vmatrix}$$ = -4, $$C_{32}$$ = -$$\begin{vmatrix} 1 & 1 \\ 2 & -3 \end{vmatrix}$$ = 5,  $$C_{33}$$ = $$\begin{vmatrix} 1 & 1 \\ 2 & 1 \end{vmatrix}$$ = -1

$$\therefore$$   adj A = $${\begin{bmatrix} 9 & -3 & 5 \\ -1 & 4 & -3 \\ -4 & 5 & -1 \end{bmatrix}}^T$$ = $$\begin{bmatrix} 9 & -1 & -4 \\ -3 & 4 & 5 \\ 5 & -3 & -1 \end{bmatrix}$$

Note : Let A be a square matrix of order n. Then, A (adj A) = |A| $$I_n$$ = (adj A) A.