Answer:
The co-factors and their values are shown in the table below.
Step-by-step explanation:
We are given the matrix, [tex]\begin{bmatrix}3&7&1\\7&1&-3\\8&5&1\end{bmatrix}[/tex].
It is required to match the co-factors with the corresponding values.
As, the co-factors are given by,
[tex]A_{c_{ij}}[/tex] = [tex](-1)^{i+j}(d)[/tex], where the d= determinant of the matrix after removing the i- row and j- column.
So, we have,
1. [tex]A_{c_{11}}[/tex].
So, [tex]A_{c_{11}}=(-1)^{1+1}(1\times 1-5\times (-3))[/tex]
i.e. [tex]A_{c_{11}}=(-1)^{2}(1+15)=16[/tex]
2. [tex]A_{c_{13}}[/tex].
So, [tex]A_{c_{13}}=(-1)^{1+3}(7\times 5-1\times 8)[/tex]
i.e. [tex]A_{c_{13}}=(-1)^{4}(35-8)=27[/tex]
3. [tex]A_{c_{21}}[/tex].
So, [tex]A_{c_{21}}=(-1)^{2+1}(7\times 1-5\times 1)[/tex]
i.e. [tex]A_{c_{21}}=(-1)^{3}(7-5)=-2[/tex]
4. [tex]A_{c_{22}}[/tex].
So, [tex]A_{c_{22}}=(-1)^{2+2}(3\times 1-8\times 1)[/tex]
i.e. [tex]A_{c_{22}}=(-1)^{4}(3-8)=-5[/tex]
5. [tex]A_{c_{31}}[/tex].
So, [tex]A_{c_{31}}=(-1)^{3+1}(7\times (-3)-1\times 1)[/tex]
i.e. [tex]A_{c_{31}}=(-1)^{4}(-21-1)=-22[/tex]
Thus, we get,
Co-factor Value
[tex]A_{c_{11}}[/tex] 16
[tex]A_{c_{13}}[/tex] 27
[tex]A_{c_{21}}[/tex] -2
[tex]A_{c_{22}}[/tex] -5
[tex]A_{c_{31}}[/tex] -22
