how to find cofactor matrix

Each element which is associated with a 2*2 determinant then the values of that determinant are called cofactors. The cofactor is defined the signed minor. An (i,j) cofactor is computed by multiplying (i,j) minor by $(-1)^{i+j}$ and is denoted by $C_{ij}$ . The formula to find cofactor = $C_{ij}=(-1)^{i+j}.M_{ij}$ where $M_{ij}$ denotes the minor of $i^{th}$ row and $j^{th}$ column of a matrix.

Co-factor of 2×2 order matrix

Let A be a square matrix. By cofactor $C_{ij}$ of an element $a_{ij}$ of A, we mean minor of $a_{ij}$ with a positive or negative sign depending on i and j. For a 2*2 matrix, negative sign is to be given the minor element $a_{12}$ and $a_{21}$ = $\begin {bmatrix} + &- \\ -&+ \end {bmatrix}$

Example 1:Consider the matrix $C=\begin {bmatrix} 5 &-1 \\ 2&2 \end {bmatrix}$

Solution:The minor of 5 is 2 and Cofactor 5 is 2 (sign unchanged)

The minor of -1 is 2 and Cofactor -1 is -2 (sign changed)

The minor of 2 is -1 and Cofactor -1 is +1 (sign changed)

The minor of 2 is 5 and Cofactor 2 is 5 (sign unchanged)

Co- factor of $C=\begin {bmatrix} 2 &-2 \\ 1&5 \end {bmatrix}$

Example 2: Consider the matrix $A=\begin {bmatrix} 5 &-3 \\ -2&0 \end {bmatrix}$

Solution: The minor of 5 is 0 and Cofactor 5 is 0 (sign unchanged)

The minor of -3 is -2 and Cofactor -3 is +2 (sign changed)

The minor of -2 is -3 and Cofactor -2 is +3 (sign changed)

The minor of 0 is 5 and Cofactor 0 is 5 (sign unchanged)

Co- factor of $A=\begin {bmatrix} 0 &2 \\ 3&5 \end {bmatrix}$

Co-factor of 3×3 order matrix

For a 3*3 matrix, negative sign is to given to minor of element : $\begin {bmatrix} + &- &+ \\ -&+&- \\ +& -&+ \end {bmatrix}$

Example 3: Consider the matrix $A=\begin {bmatrix} 2 &-3 &-1 \\ 6&4&1 \\ 0&5&3 \end {bmatrix}$

Solution: Minor of 2 is 7 and Cofactor is 7.

Minor of -3 is 18 and Cofactor is -18 (sign changed)

Minor of -1 is 30 and Cofactor are 30.

Minor of 6 is 1 and Cofactor is -1 (sign changed)

Minor of 4 is 6 and Cofactor are 6.

Minor of 1 is 10 and Cofactor is -10 (sign changed)

Minor of 0 is 1 and Cofactor are 1.

Minor of 6 is 8 and Cofactor is -8 (sign changed)

Minor of 3 is 26 and Cofactor is 26

Example 4:Consider the matrix $D=\begin {bmatrix} 3 &-2 &-1 \\ 2&1&5 \\ 0&6&4 \end {bmatrix}$

Solution:Minor of 3 is -26 and Cofactor is -26.

Minor of -2 is 18 and Cofactor is -8 (sign changed)

Minor of -1 is 12 and Cofactor is 12.

Minor of 2 is -2 and Cofactor is -2 (sign changed)

Minor of 1 is 12 and Cofactor are 12.

Minor of 5 is 18 and Cofactor is -18 (sign changed)

Minor of 0 is -9 and Cofactor are -9.

Minor of 6 is 17 and Cofactor is -17 (sign changed)

Minor of 4 is 7 and Cofactor are 7.

Exercise

Find the co-factors of the matrix $A=\begin {bmatrix} -1 &-2 &-2 \\ 2&1&-2 \\ 2&-2&1 \end {bmatrix}$ .
Find the co-factors of matrix $F=\begin {bmatrix} 4 &2 &3 \\ 4&0&1 \\ 1&1&0 \end {bmatrix}$ .
Find the co-factors of matrix $A=\begin {bmatrix} 8 &-9 \\ -5&6 \end {bmatrix}$ .
Find the co-factors of matrix $D=\begin {bmatrix} 4 &-5 \\ 2&1 \end {bmatrix}$ .
Find the co-factors of the matrix $G=\begin {bmatrix} 3 &-4 &1 \\ -3&6&-1 \\ 4&-6&2 \end {bmatrix}$ .