Let A=(1, 2; -2, -5). Let α, β ∈ ℝ be such that α A²+β A=2 I. Then α+β is equal to

Solution

Maths sir · 5 steps

Do not chase α and β by brute force. Cayley-Hamilton hands you the relation between A² and A directly.

1. Trace and determinant. For A, the trace is 1 + (-5) = -4 and the determinant is (1)(-5) - (2)(-2) = -5 + 4 = -1.
2. Cayley-Hamilton. Every 2 × 2 matrix satisfies A² - (tr A)A + (det A)I = O. Putting in the values gives the equation below.

   A² + 4A - I = O ⇒ A² + 4A = I

3. Check it. A² = (-3, -8; 8, 21) and 4A = (4, 8; -8, -20). Their sum is (1, 0; 0, 1) = I.
4. Match the form. Multiply by 2 to get 2A² + 8A = 2I, so α = 2 and β = 8. These values are unique: substituting A² = I - 4A gives α I + (β - 4α)A = 2I. Since A is not a multiple of I, this forces α = 2 and β = 4α = 8.
5. So. α + β = 2 + 8 = 10, which is option (d).

The idea to keep. Any equation linking A², A and I for a 2 × 2 matrix comes from Cayley-Hamilton: A² - (tr A)A + (det A)I = O. Find trace and determinant first.

Answer from the NTA final answer key. Solution by PrepWiser.