To determine which matrices [tex]\( A \)[/tex] satisfy the condition that [tex]\( A^2 \)[/tex] has identical diagonal elements, let's step through the process of squaring each matrix and evaluating its diagonal elements.
To clarify, for each given matrix [tex]\( A \)[/tex], we will:
1. Calculate [tex]\( A^2 \)[/tex].
2. Check the diagonal elements of [tex]\( A^2 \)[/tex].
3. Determine if all diagonal elements of [tex]\( A^2 \)[/tex] are identical.
Given matrices:
[tex]\[
A_1 = \left[\begin{array}{lll}
7 & 1 & 5 \\
1 & 5 & 7 \\
5 & 7 & 1
\end{array}\right]
\][/tex]
[tex]\[
A_2 = \left[\begin{array}{ccc}
7 & -1 & 5 \\
-1 & 7 & 6 \\
5 & 4 & 5
\end{array}\right]
\][/tex]
[tex]\[
A_3 = \left[\begin{array}{lll}
7 & 4 & 6 \\
6 & 4 & 7 \\
4 & 6 & 7
\end{array}\right]
\][/tex]
[tex]\[
A_4 = \left[\begin{array}{ccc}
9 & 18 & 27 \\
27 & -9 & 18 \\
18 & 27 & 9
\end{array}\right]
\][/tex]
[tex]\[
A_5 = \left[\begin{array}{lll}
8 & 1 & 6 \\
6 & 8 & 1 \\
1 & 6 & 8
\end{array}\right]
\][/tex]
[tex]\[
A_6 = \left[\begin{array}{ccc}
4 & 5 & 6 \\
6 & 4 & -5 \\
5 & 6 & 4
\end{array}\right]
\][/tex]
Now we determine the matrices for which [tex]\( A^2 \)[/tex] has identical diagonal elements:
1. Matrix [tex]\( A_1 \)[/tex]:
[tex]\[
A_1 = \left[\begin{array}{lll}
7 & 1 & 5 \\
1 & 5 & 7 \\
5 & 7 & 1
\end{array}\right]
\][/tex]
[tex]\[
A_1^2 = \left[\begin{array}{lll}
55 & 47 & 42 \\
47 & 75 & 12 \\
42 & 12 & 75
\end{array}\right]
\][/tex]
Diagonal elements: [tex]\( 55, 75, 75 \)[/tex]
Here, not all diagonal elements are identical. Hence, [tex]\( A_1 \)[/tex] is not the correct matrix.
2. Matrix [tex]\( A_2 \)[/tex]:
[tex]\[
A_2 = \left[\begin{array}{ccc}
7 & -1 & 5 \\
-1 & 7 & 6 \\
5 & 4 & 5
\end{array}\right]
\][/tex]
[tex]\[
A_2^2 = \left[\begin{array}{ccc}
75 & 25 & 60 \\
25 & 86 & 14 \\
60 & 14 & 70
\end{array}\right]
\][/tex]
Diagonal elements: [tex]\( 75, 86, 70 \)[/tex]
Diagonal elements are not identical for [tex]\( A_2 \)[/tex].
3. Matrix [tex]\( A_3 \)[/tex]:
[tex]\[
A_3 = \left[\begin{array}{lll}
7 & 4 & 6 \\
6 & 4 & 7 \\
4 & 6 & 7
\end{array}\right]
\][/tex]
[tex]\[
A_3^2 = \left[\begin{array}{lll}
101 & 82 & 92 \\
82 & 101 & 92 \\
92 & 92 & 101
\end{array}\right]
\][/tex]
Diagonal elements: [tex]\( 101, 101, 101 \)[/tex]
All diagonal elements are identical for [tex]\( A_3 \)[/tex].
4. Matrix [tex]\( A_4 \)[/tex]:
[tex]\[
A_4 = \left[\begin{array}{ccc}
9 & 18 & 27 \\
27 & -9 & 18 \\
18 & 27 & 9
\end{array}\right]
\][/tex]
[tex]\[
A_4^2 = \left[\begin{array}{ccc}
1188 & 540 & 972 \\
540 & 1188 & 972 \\
972 & 972 & 1188
\end{array}\right]
\][/tex]
Diagonal elements: [tex]\( 1188, 1188, 1188 \)[/tex]
All diagonal elements are identical for [tex]\( A_4 \)[/tex].
5. Matrix [tex]\( A_5 \)[/tex]:
[tex]\[
A_5 = \left[\begin{array}{lll}
8 & 1 & 6 \\
6 & 8 & 1 \\
1 & 6 & 8
\end{array}\right]
\][/tex]
[tex]\[
A_5^2 = \left[\begin{array}{lll}
101 & 62 & 96 \\
62 & 101 & 96 \\
96 & 96 & 101
\end{array}\right]
\][/tex]
Diagonal elements: [tex]\( 101, 101, 101 \)[/tex]
6. Matrix [tex]\( A_6 \)[/tex]:
[tex]\[
A_6 = \left[\begin{array}{ccc}
4 & 5 & 6 \\
6 & 4 & -5 \\
5 & 6 & 4
\end{array}\right]
\][/tex]
[tex]\[
A_6^2 = \left[\begin{array}{ccc}
97 & 22 & 29 \\
22 & 97 & 85 \\
29 & 85 & 97
\end{array}\right]
\][/tex]
Diagonal elements: [tex]\( 97, 97, 97 \)[/tex]
Based on this evaluation, the matrices [tex]\( A \)[/tex] such that [tex]\( A^2 \)[/tex] has identical diagonal elements are [tex]\( A_3 \)[/tex], [tex]\( A_4 \)[/tex], and [tex]\( A_5 \)[/tex], thus the correct indexes of the matrices are:
[tex]\[ [0, 3, 4] \][/tex]
