Question

Match the cofactors to their corresponding entries in the matrix.

Answer 1

The co-factors and their values are shown in the table below.

Explanation:

We are given the matrix,
`\begin{bmatrix}3&7&1\\7&1&-3\\8&5&1\end{bmatrix}`.

It is required to match the co-factors with the corresponding values.

As, the co-factors are given by,

`A_{c_(ij)}` =
`(-1)^(i+j)(d)`, where the d= determinant of the matrix after removing the i- row and j- column.

So, we have,

1.
`A_{c_(11)}`.

So,
`A_{c_(11)}=(-1)^(1+1)(1* 1-5* (-3))`

i.e.
`A_{c_(11)}=(-1)^(2)(1+15)=16`

2.
`A_{c_(13)}`.

So,
`A_{c_(13)}=(-1)^(1+3)(7* 5-1* 8)`

i.e.
`A_{c_(13)}=(-1)^(4)(35-8)=27`

3.
`A_{c_(21)}`.

So,
`A_{c_(21)}=(-1)^(2+1)(7* 1-5* 1)`

i.e.
`A_{c_(21)}=(-1)^(3)(7-5)=-2`

4.
`A_{c_(22)}`.

So,
`A_{c_(22)}=(-1)^(2+2)(3* 1-8* 1)`

i.e.
`A_{c_(22)}=(-1)^(4)(3-8)=-5`

5.
`A_{c_(31)}`.

So,
`A_{c_(31)}=(-1)^(3+1)(7* (-3)-1* 1)`

i.e.
`A_{c_(31)}=(-1)^(4)(-21-1)=-22`

Thus, we get,

Co-factor Value

`A_{c_(11)}` 16

`A_{c_(13)}` 27

`A_{c_(21)}` -2

`A_{c_(22)}` -5

`A_{c_(31)}` -22