Additional Cofactor Questions with Answers

11. Find the cofactor of element 4 in the matrix:

[ A= \begin{bmatrix} 1 & 2 & 3
4 & 5 & 6
7 & 8 & 9 \end{bmatrix} ]

Answer:

Element 4 is at position ((2,1)).

Delete row 2 and column 1:

[ \begin{bmatrix} 2 & 3
8 & 9 \end{bmatrix} ]

Minor:

[ M_{21}=(2)(9)-(3)(8) ]

[ =18-24=-6 ]

Cofactor:

[ C_{21}=(-1)^{2+1}(-6) ]

[ =(-1)^3(-6)=6 ]

Final Answer: 6

12. Find the cofactor of element 5 in the matrix:

[ A= \begin{bmatrix} 2 & 1 & 4
3 & 5 & 6
7 & 8 & 9 \end{bmatrix} ]

Answer:

Element 5 is at position ((2,2)).

Delete row 2 and column 2:

[ \begin{bmatrix} 2 & 4
7 & 9 \end{bmatrix} ]

Minor:

[ M_{22}=(2)(9)-(4)(7) ]

[ =18-28=-10 ]

Cofactor:

[ C_{22}=(-1)^{2+2}(-10) ]

[ =(-10) ]

Final Answer: (-10)

13. Find the cofactor of element 3 in the matrix:

[ A= \begin{bmatrix} 3 & 1 & 2
4 & 5 & 6
7 & 8 & 9 \end{bmatrix} ]

Answer:

Element 3 is at position ((1,1)).

Delete row 1 and column 1:

[ \begin{bmatrix} 5 & 6
8 & 9 \end{bmatrix} ]

Minor:

[ M_{11}=(5)(9)-(6)(8) ]

[ =45-48=-3 ]

Cofactor:

[ C_{11}=(-1)^{1+1}(-3) ]

[ =-3 ]

Final Answer: (-3)

14. Find the cofactor of element 2 in the matrix:

[ A= \begin{bmatrix} 1 & 2 & 0
4 & 5 & 6
7 & 8 & 9 \end{bmatrix} ]

Answer:

Element 2 is at position ((1,2)).

Delete row 1 and column 2:

[ \begin{bmatrix} 4 & 6
7 & 9 \end{bmatrix} ]

Minor:

[ M_{12}=(4)(9)-(6)(7) ]

[ =36-42=-6 ]

Cofactor:

[ C_{12}=(-1)^{1+2}(-6) ]

[ =(-1)^3(-6)=6 ]

Final Answer: 6

15. Find the cofactor matrix of:

[ A= \begin{bmatrix} 1 & 2 & 3
0 & 4 & 5
1 & 0 & 6 \end{bmatrix} ]

Answer:

Cofactors:

[ C_{11}= \begin{vmatrix} 4 & 5
0 & 6 \end{vmatrix} =24 ]

[ C_{12}=- \begin{vmatrix} 0 & 5
1 & 6 \end{vmatrix} =-(0-5)=5 ]

[ C_{13}= \begin{vmatrix} 0 & 4
1 & 0 \end{vmatrix} =(0-4)=-4 ]

[ C_{21}=- \begin{vmatrix} 2 & 3
0 & 6 \end{vmatrix} =-(12)=-12 ]

[ C_{22}= \begin{vmatrix} 1 & 3
1 & 6 \end{vmatrix} =6-3=3 ]

[ C_{23}=- \begin{vmatrix} 1 & 2
1 & 0 \end{vmatrix} =-(0-2)=2 ]

[ C_{31}= \begin{vmatrix} 2 & 3
4 & 5 \end{vmatrix} =10-12=-2 ]

[ C_{32}=- \begin{vmatrix} 1 & 3
0 & 5 \end{vmatrix} =-(5)=-5 ]

[ C_{33}= \begin{vmatrix} 1 & 2
0 & 4 \end{vmatrix} =4 ]

Cofactor Matrix:

[ \begin{bmatrix} 24 & 5 & -4
-12 & 3 & 2
-2 & -5 & 4 \end{bmatrix} ]

16. State the formula for cofactor.

Answer:

[ C_{ij}=(-1)^{i+j}M_{ij} ]

where:

• (C_{ij}) = cofactor
• (M_{ij}) = minor
• (i,j) = row and column positions

17. Find the cofactor of element 7 in the matrix:

[ A= \begin{bmatrix} 2 & 1 & 3
4 & 5 & 6
7 & 8 & 1 \end{bmatrix} ]

Answer:

Element 7 is at position ((3,1)).

Delete row 3 and column 1:

[ \begin{bmatrix} 1 & 3
5 & 6 \end{bmatrix} ]

Minor:

[ M_{31}=(1)(6)-(3)(5) ]

[ =6-15=-9 ]

Cofactor:

[ C_{31}=(-1)^{3+1}(-9) ]

[ =(-9) ]

Final Answer: (-9)

18. Find the cofactor of element 6 in the matrix:

[ A= \begin{bmatrix} 1 & 2 & 3
4 & 5 & 6
0 & 1 & 2 \end{bmatrix} ]

Answer:

Element 6 is at position ((2,3)).

Delete row 2 and column 3:

[ \begin{bmatrix} 1 & 2
0 & 1 \end{bmatrix} ]

Minor:

[ M_{23}=(1)(1)-(2)(0)=1 ]

Cofactor:

[ C_{23}=(-1)^{2+3}(1) ]

[ =-1 ]

Final Answer: (-1)

19. What is a cofactor matrix?

Answer:

A cofactor matrix is the matrix formed by replacing every element of a matrix with its corresponding cofactor.

20. How is adjoint related to cofactor matrix?

Answer:

The adjoint (adjugate) of a matrix is obtained by taking the transpose of the cofactor matrix.

[ \text{Adj}(A)=(\text{Cofactor Matrix})^T ]