# Minors and Cofactors of a Matrix (3×3 and 2×2) with Examples

Here you will learn how to find minors and cofactors of a matrix of order 3×3 and 2×2 with examples.

Let’s begin –

## How to find Minors and Cofactors of a Matrix

### Minor of Matrix (3×3 and 2×2)

Let A = $$[a_{ij}]$$ be a square matrix of order n. The minor $$M_{ij}$$ of $$a_{ij}$$ in A is the determinant of the square sub-matrix of order (n – 1) obtained by leaving $$i^{ith}$$ row and $$j^{ith}$$ column of A.

Example : if A = $$\begin{bmatrix} 4 & -7 \\ -3 & 2 \end{bmatrix}$$, then

$$M_{11}$$ = Minor of  $$a_{11}$$ = 2,  $$M_{12}$$ = Minor of  $$a_{12}$$ = -3,

$$M_{21}$$ = Minor of  $$a_{21}$$ = -7,  $$M_{22}$$ = Minor of  $$a_{22}$$ = 4

If A = $$\begin{bmatrix} 1 & 2 & 3 \\ -3 & 2 & -1 \\ 2 & -4 & 3 \end{bmatrix}$$, then

$$M_{11}$$ = Minor of  $$a_{11}$$

$$\implies$$ $$M_{11}$$ = Determinant of the 2×2 square sub matrix obtained by leaving first row and first column of A

$$\implies$$ $$M_{11}$$ = $$\begin{vmatrix} 2 & -1 \\ -4 & 3 \end{vmatrix}$$

Similarly, we obtain

$$M_{12}$$ = Minor of  $$a_{12}$$ = $$\begin{vmatrix} -3 & -1 \\ 2 & 3 \end{vmatrix}$$ = -7

$$M_{13}$$ = Minor of  $$a_{13}$$ = $$\begin{vmatrix} -3 & 2 \\ 2 & -4 \end{vmatrix}$$ = 8

$$M_{21}$$ = Minor of  $$a_{21}$$ = $$\begin{vmatrix} 2 & 3 \\ -4 & 3 \end{vmatrix}$$ = 18

$$M_{22}$$ = Minor of  $$a_{22}$$ = $$\begin{vmatrix} 1 & 3 \\ 2 & 3 \end{vmatrix}$$ = -3    etc.

### Cofactor of Matrix (3×3 and 2×2)

Let A = $$[a_{ij}]$$ be a square matrix of order n. The cofactor $$C_{ij}$$ of $$a_{ij}$$ in A is equal to $$(-1)^{i+j}$$ times the determinant of the sub-matrix of order (n – 1) obtained by leaving $$i^{ith}$$ row and $$j^{ith}$$ column of A.

It follows from this definition that

$$C_{ij}$$ = Cofactor of $$a_{ij}$$ in A = $$(-1)^{i+j}$$$$M_{ij}$$, where $$M_{ij}$$ is minor of $$a_{ij}$$ in A.

Thus we have

$$C_{ij}$$ = $$M_{ij}$$ if i + j  is even

$$C_{ij}$$ = $$-M_{ij}$$ if i + j  is odd

Example : if A = $$\begin{bmatrix} 4 & -7 \\ -3 & 2 \end{bmatrix}$$, then

$$C_{11}$$ = $$(-1)^{1+1}$$$$M_{11}$$ = $$M_{11}$$  = 2, $$C_{12}$$ = $$(-1)^{1+2}$$$$M_{12}$$ = -$$M_{12}$$  = -(-3) = 3,

$$C_{21}$$ = $$(-1)^{2+1}$$$$M_{21}$$ = $$M_{21}$$  = -(-7) = 7, $$C_{22}$$ = $$(-1)^{2+2}$$$$M_{22}$$ = $$M_{22}$$  = 4

If A = $$\begin{bmatrix} 1 & 2 & 3 \\ -3 & 2 & -1 \\ 2 & -4 & 3 \end{bmatrix}$$, then

$$C_{11}$$ = $$(-1)^{1+1}$$$$M_{11}$$ = $$M_{11}$$ = $$\begin{vmatrix} 2 & -1 \\ -4 & 3 \end{vmatrix}$$ = 2

$$C_{12}$$ = $$(-1)^{1+2}$$$$M_{12}$$ = -$$M_{12}$$ = -$$\begin{vmatrix} -3 & -1 \\ 2 & 3 \end{vmatrix}$$ = 7

$$C_{13}$$ = $$(-1)^{1+3}$$$$M_{13}$$ = $$M_{13}$$ = $$\begin{vmatrix} -3 & 2 \\ 2 & -4 \end{vmatrix}$$ = 8

$$C_{23}$$ = $$(-1)^{2+3}$$$$M_{23}$$ = -$$M_{23}$$ = -$$\begin{vmatrix} 1 & 2 \\ 2 & -4 \end{vmatrix}$$ = 8    etc.