To solve this problem, we need to understand the given polynomial equation for the matrix \( A \) and use it to find an expression for \( A^5 \), ultimately aiming to find the values of \( l \) and \( m \) such that \( A^5 - 6A^4 + 12A^3 - 6A^2 + 2A + 2I = lA + mI \).
We start with the given equation:
\(A^3 - 5A^2 + 7A + I = 0.\)
This implies:
\(A^3 = 5A^2 - 7A - I.\)
Next, we express higher powers of \( A \) using this result. Let's find \( A^4 \) and \( A^5 \):
Calculate \( A^4 \):
\(A^4 = A \cdot A^3 = A(5A^2 - 7A - I).\)
Simplifying, we get:
\(A^4 = 5A^3 - 7A^2 - A.\)
Substitute \( A^3 = 5A^2 - 7A - I \) into this equation:
\(A^4 = 5(5A^2 - 7A - I) - 7A^2 - A.\)
\(A^4 = 25A^2 - 35A - 5I - 7A^2 - A.\)
\(A^4 = 18A^2 - 36A - 5I.\)
Calculate \( A^5 \):
\(A^5 = A \cdot A^4 = A(18A^2 - 36A - 5I).\)
Simplifying, we get:
\(A^5 = 18A^3 - 36A^2 - 5A.\)
Substitute \( A^3 = 5A^2 - 7A - I \) into this equation:
\(A^5 = 18(5A^2 - 7A - I) - 36A^2 - 5A.\)
\(A^5 = 90A^2 - 126A - 18I - 36A^2 - 5A.\)
\(A^5 = 54A^2 - 131A - 18I.\)
Substitute the expressions for \( A^5 \) and simplify:
The given equation:
\(A^5 - 6A^4 + 12A^3 - 6A^2 + 2A + 2I = lA + mI.\)
Substitute known values of \( A^4 \) and \( A^3 \):
\((54A^2 - 131A - 18I) - 6(18A^2 - 36A - 5I) + 12(5A^2 - 7A - I) - 6A^2 + 2A + 2I = lA + mI.\)
Calculate each expression:
\(= 54A^2 - 131A - 18I - (108A^2 - 216A - 30I) + (60A^2 - 84A - 12I) - 6A^2 + 2A + 2I.\)
Simplify the expression:
\(= (54A^2 - 108A^2 + 60A^2 - 6A^2) + (-131A + 216A - 84A + 2A) + (-18I + 30I - 12I + 2I).\)
\(= 0A^2 + 3A + 2I.\)
Therefore, \( l = 3 \) and \( m = 2 \). Thus, \( l + m = 3 + 2 = 5 \).
Hence, the correct answer is 5.