Let A be a 3x3 matrix such that A² - A + 7I = 0. Statement I: A⁵ = 17(5I - A). Statement II: The polynomial A³ - A² + A + I can be reduced to 5(A - I). Then: Both the statements are true, Statement - I is true, but Statement - II is false, Both the statements are false, Statement - I is false, but Statement - II is true

Given A² - A + 7I = 0

A² = A - 7I
A⁵ = A³A² = A³(A - 7I) = A⁴ - 7A³ = A²(A²)-7A³ = (A-7I)(A-7I)-7A³ = A²-14A+49I-7A³ = (A-7I)-14A+49I-7A³ = -13A+56I-7A³

From A² - A + 7I = 0, we have A² = A - 7I
Multiplying by A, we get A³ = A² - 7A = (A - 7I) - 7A = -6A - 7I
Then A⁴ = A(-6A - 7I) = -6A² - 7A = -6(A - 7I) - 7A = -13A + 42I
A⁵ = A(-13A + 42I) = -13A² + 42A = -13(A - 7I) + 42A = 29A + 91I

Let's check statement I:
A⁵ = 17(5I - A)
29A + 91I = 85I - 17A
46A = -8I
This is not true in general.

Let's use A² = A - 7I
A³ = A(A - 7I) = A² - 7A = (A - 7I) - 7A = -6A - 7I
A³ - A² + A + I = (-6A - 7I) - (A - 7I) + A + I = -6A - 7I - A + 7I + A + I = -6A + I ≠ 5(A - I)
Statement II is false.
Therefore, Statement I is false, but Statement II is false.