Alright, we'll be ordering the entries of the given matrix [tex]$ A $[/tex] based on their cofactor values. We have the cofactor values and their sorted order. Let's match these cofactor values back to their corresponding matrix entries.
The given matrix [tex]$ A $[/tex] is:
[tex]\[
\left[\begin{array}{ccc}
7 & 5 & 3 \\
-7 & 4 & -1 \\
-8 & 2 & 1
\end{array}\right]
\][/tex]
The cofactor values we have for the corresponding entries (in flat form row-wise) are:
[tex]\[ [6, 15, 18, 1, 31, -54, -17, -14, 63] \][/tex]
The sorted cofactor values are:
[tex]\[ [-54, -17, -14, 1, 6, 15, 18, 31, 63] \][/tex]
Now, let's link each cofactor value to its corresponding matrix entry:
1. Entry with cofactor -54: The cofactor -54 corresponds to the entry (2, 1) which is -7.
2. Entry with cofactor -17: The cofactor -17 corresponds to the entry (3, 1) which is -8.
3. Entry with cofactor -14: The cofactor -14 corresponds to the entry (3, 2) which is 2.
4. Entry with cofactor 1: The cofactor 1 corresponds to the entry (2, 2) which is 4.
5. Entry with cofactor 6: The cofactor 6 corresponds to the entry (1, 1) which is 7.
6. Entry with cofactor 15: The cofactor 15 corresponds to the entry (1, 2) which is 5.
7. Entry with cofactor 18: The cofactor 18 corresponds to the entry (1, 3) which is 3.
8. Entry with cofactor 31: The cofactor 31 corresponds to the entry (2, 3) which is -1.
9. Entry with cofactor 63: The cofactor 63 corresponds to the entry (3, 3) which is 1.
Let's arrange the entries of the matrix in increasing order of their cofactor values:
[tex]\[
[-7, -8, 2, 4, 7, 5, 3, -1, 1]
\][/tex]
Therefore, the entries of matrix [tex]$ A $[/tex] in increasing order of their cofactor values are:
- [tex]\( -7 \)[/tex]
- [tex]\( -8 \)[/tex]
- [tex]\( 2 \)[/tex]
- [tex]\( 4 \)[/tex]
- [tex]\( 7 \)[/tex]
- [tex]\( 5 \)[/tex]
- [tex]\( 3 \)[/tex]
- [tex]\( -1 \)[/tex]
- [tex]\( 1 \)[/tex]
