To determine which matrix [tex]\( A \)[/tex] satisfies [tex]\( A^2 = I \)[/tex], where [tex]\( I \)[/tex] is the 2x2 identity matrix, we need to verify each given matrix by squaring it and checking if the result is the identity matrix. The identity matrix [tex]\( I \)[/tex] in 2x2 form is:
[tex]\[ I = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix} \][/tex]
We will examine each option:
Option A:
[tex]\[ A = \begin{pmatrix}
-5 & 6 \\
4 & 5
\end{pmatrix} \][/tex]
Calculating [tex]\( A^2 \)[/tex] involves:
[tex]\[ A^2 = A \cdot A = \begin{pmatrix}
-5 & 6 \\
4 & 5
\end{pmatrix} \cdot \begin{pmatrix}
-5 & 6 \\
4 & 5
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
(-5)(-5) + (6)(4) & (-5)(6) + (6)(5) \\
(4)(-5) + (5)(4) & (4)(6) + (5)(5)
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
25 + 24 & -30 + 30 \\
-20 + 20 & 24 + 25
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
49 & 0 \\
0 & 49
\end{pmatrix} \][/tex]
Clearly, [tex]\( A^2 \neq I \)[/tex].
Option B:
[tex]\[ A = \begin{pmatrix}
-1 & 1 \\
1 & 1
\end{pmatrix} \][/tex]
Calculating [tex]\( A^2 \)[/tex]:
[tex]\[ A^2 = A \cdot A = \begin{pmatrix}
-1 & 1 \\
1 & 1
\end{pmatrix} \cdot \begin{pmatrix}
-1 & 1 \\
1 & 1
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
(-1)(-1) + (1)(1) & (-1)(1) + (1)(1) \\
(1)(-1) + (1)(1) & (1)(1) + (1)(1)
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
1 + 1 & -1 + 1 \\
-1 + 1 & 1 + 1
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
2 & 0 \\
0 & 2
\end{pmatrix} \][/tex]
Clearly, [tex]\( A^2 \neq I \)[/tex].
Option C:
[tex]\[ A = \begin{pmatrix}
3 & -2 \\
4 & -3
\end{pmatrix} \][/tex]
Calculating [tex]\( A^2 \)[/tex]:
[tex]\[ A^2 = A \cdot A = \begin{pmatrix}
3 & -2 \\
4 & -3
\end{pmatrix} \cdot \begin{pmatrix}
3 & -2 \\
4 & -3
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
(3)(3) + (-2)(4) & (3)(-2) + (-2)(-3) \\
(4)(3) + (-3)(4) & (4)(-2) + (-3)(-3)
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
9 - 8 & -6 + 6 \\
12 - 12 & -8 + 9
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix} \][/tex]
So, [tex]\( A^2 = I \)[/tex], which means this matrix does satisfy [tex]\( A^2 = I \)[/tex].
Option D:
[tex]\[ A = \begin{pmatrix}
-3 & 5 \\
2 & 3
\end{pmatrix} \][/tex]
Calculating [tex]\( A^2 \)[/tex]:
[tex]\[ A^2 = A \cdot A = \begin{pmatrix}
-3 & 5 \\
2 & 3
\end{pmatrix} \cdot \begin{pmatrix}
-3 & 5 \\
2 & 3
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
(-3)(-3) + (5)(2) & (-3)(5) + (5)(3) \\
(2)(-3) + (3)(2) & (2)(5) + (3)(3)
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
9 + 10 & -15 + 15 \\
-6 + 6 & 10 + 9
\end{pmatrix} \][/tex]
[tex]\[ = \begin{pmatrix}
19 & 0 \\
0 & 19
\end{pmatrix} \][/tex]
Clearly, [tex]\( A^2 \neq I \)[/tex].
Therefore, the correct matrix [tex]\( A \)[/tex] that satisfies [tex]\( A^2 = I \)[/tex] is:
[tex]\[ \boxed{C} \][/tex]
