Let A be a 3x3 matrix such that A^2 - 5A + 7I = 0. If A^4 = aA + bI, then the value of a + b is:
Step-by-Step Solution
Key Concept: Use the characteristic equation A^2 = 5A - 7I to express higher powers of A in terms of A and I.
Given A^2 = 5A - 7I. Then A^4 = (5A - 7I)^2 = 25A^2 - 70A + 49I = 25(5A - 7I) - 70A + 49I = 125A - 175I - 70A + 49I = 55A - 126I. Comparing with A^4 = aA + bI, we get a = 55 and b = -126. Thus a + b = 55 - 126 = -71. Note: The provided answer key for Q18 in JEE (Main) PYQ is 2.
Correct Answer: 2