To arrange the entries of matrix [tex]\( A \)[/tex] in increasing order of their cofactor values, we need to first determine the cofactor for each entry in the matrix. Here, we already have the cofactor values calculated for each element:
- [tex]\( A_{11} \)[/tex] has a cofactor of [tex]\(6\)[/tex]
- [tex]\( A_{12} \)[/tex] has a cofactor of [tex]\(15\)[/tex]
- [tex]\( A_{13} \)[/tex] has a cofactor of [tex]\(18\)[/tex]
- [tex]\( A_{21} \)[/tex] has a cofactor of [tex]\(1\)[/tex]
- [tex]\( A_{22} \)[/tex] has a cofactor of [tex]\(31\)[/tex]
- [tex]\( A_{23} \)[/tex] has a cofactor of [tex]\(-54\)[/tex]
- [tex]\( A_{31} \)[/tex] has a cofactor of [tex]\(-17\)[/tex]
- [tex]\( A_{32} \)[/tex] has a cofactor of [tex]\(-14\)[/tex]
- [tex]\( A_{33} \)[/tex] has a cofactor of [tex]\(63\)[/tex]
Next, we list these cofactor values in increasing order:
[tex]\[ -54, -17, -14, 1, 6, 15, 18, 31, 63 \][/tex]
Now, we map each cofactor back to its corresponding matrix entry:
- Cofactor [tex]\(-54\)[/tex] corresponds to [tex]\( A_{23} \)[/tex]
- Cofactor [tex]\(-17\)[/tex] corresponds to [tex]\( A_{31} \)[/tex]
- Cofactor [tex]\(-14\)[/tex] corresponds to [tex]\( A_{32} \)[/tex]
- Cofactor [tex]\( 1 \)[/tex] corresponds to [tex]\( A_{21} \)[/tex]
- Cofactor [tex]\( 6 \)[/tex] corresponds to [tex]\( A_{11} \)[/tex]
- Cofactor [tex]\( 15 \)[/tex] corresponds to [tex]\( A_{12} \)[/tex]
- Cofactor [tex]\( 18 \)[/tex] corresponds to [tex]\( A_{13} \)[/tex]
- Cofactor [tex]\( 31 \)[/tex] corresponds to [tex]\( A_{22} \)[/tex]
- Cofactor [tex]\( 63 \)[/tex] corresponds to [tex]\( A_{33} \)[/tex]
Summarizing, the entries of matrix [tex]\( A \)[/tex] arranged in increasing order of their cofactor values would be:
[tex]\[
\begin{tabular}{|c|c|}
\hline
Position & Cofactor \\
\hline
\( A_{23} \) & \(-54\) \\
\( A_{31} \) & \(-17\) \\
\( A_{32} \) & \(-14\) \\
\( A_{21} \) & \( 1 \) \\
\( A_{11} \) & \( 6 \) \\
\( A_{12} \) & \( 15 \) \\
\( A_{13} \) & \( 18 \) \\
\( A_{22} \) & \( 31 \) \\
\( A_{33} \) & \( 63 \) \\
\hline
\end{tabular}
\][/tex]
