Matrices — Hard JEE math MCQ
**[JEE Mains 2026]** If A = [[2, 3], [3, 5]], then the value of det(A⁶ - 6A⁵ + 9A⁴) is:
- A. 0
- B. 1
- C. 64
- D. 81
Solution
detA. = 10 - 9 = 1.
A⁶ - 6A⁵ + 9A⁴ = A⁴(A² - 6A + 9I) = A⁴(A - 3I)².
By Cayley-Hamilton: A² - 7A + I = 0 (characteristic equation).
We need to check if (A - 3I)² relates to this.
det(A - 3I) = det([[-1, 3], [3, 2]]) = -2 - 9 = -11 ≠ 0.
Actually: A⁶ - 6A⁵ + 9A⁴ = A⁴(A - 3I)².
det = det(A)⁴ × det(A - 3I)² = 1 × 121 = 121.
Let me recalculate. The answer given suggests 0, which would mean A - 3I is singular or there's a factorization issue.