Sure, let's work through the problem step-by-step.
We have a matrix [tex]\( A \)[/tex] given by:
[tex]\[
A = \begin{pmatrix}
7 & 5 & 3 \\
-7 & 4 & -1 \\
-8 & 2 & 1
\end{pmatrix}
\][/tex]
First, we need to determine the cofactor of each element in the matrix. Here is the resulting cofactor matrix:
[tex]\[
\begin{pmatrix}
6 & 15 & 18 \\
1 & 31 & -54 \\
-17 & -14 & 63
\end{pmatrix}
\][/tex]
Now, we need to list each entry of matrix [tex]\(A\)[/tex] together with its cofactor:
- For [tex]\( 7 \)[/tex], the cofactor is [tex]\(6\)[/tex]
- For [tex]\( 5 \)[/tex], the cofactor is [tex]\(15\)[/tex]
- For [tex]\( 3 \)[/tex], the cofactor is [tex]\(18\)[/tex]
- For [tex]\(-7 \)[/tex], the cofactor is [tex]\(1\)[/tex]
- For [tex]\(4 \)[/tex], the cofactor is [tex]\(31\)[/tex]
- For [tex]\(-1 \)[/tex], the cofactor is [tex]\(-54\)[/tex]
- For [tex]\(-8 \)[/tex], the cofactor is [tex]\(-17\)[/tex]
- For [tex]\(2 \)[/tex], the cofactor is [tex]\(-14\)[/tex]
- For [tex]\(1 \)[/tex], the cofactor is [tex]\(63\)[/tex]
We now arrange the entries of matrix [tex]\( A \)[/tex] in order of their cofactor values, in ascending order:
[tex]\[
-1, -8, 2, -7, 7, 5, 3, 4, 1
\][/tex]
Hence, the ordered list of entries in the matrix [tex]\( A \)[/tex] according to increasing cofactor values is:
[tex]\[
\boxed{-1, -8, 2, -7, 7, 5, 3, 4, 1}
\][/tex]
