JEE Main 2022 · 27 July · evening · Single correct · +4 / −1

Let A=(4, -2; α, β).
If A²+γ A+18 I=O, then det(A) is equal to _____________.

  1. (a) -18
  2. (b) 18
  3. (c) -50
  4. (d) 50
Answer and solution

Answer (b) 18

Solution

Maths sir · 4 steps

Every 2 by 2 matrix satisfies its own characteristic equation, so match it term by term with the given one.

  1. Cayley-Hamilton. For any 2 × 2 matrix A, the Cayley-Hamilton theorem gives an equation that always holds. The constant term in it is det(A), with a plus sign.

    A² - (tr A) A + (det A) I = O

  2. Subtract the equations. Subtracting this from A² + γ A + 18I = O gives (γ + tr A) A = (det A - 18) I. If γ + tr A ≠ 0, then A would be a multiple of I.
  3. A is not scalar. A multiple of I has zeros off the diagonal, but A has -2 in the top-right corner. So γ + tr A = 0, which leaves (det A - 18) I = O.
  4. So. γ = -tr A and det(A) = 18, which is option (b).

The idea to keep. If a non-scalar 2 × 2 matrix satisfies A² + pA + qI = O, that equation must be its characteristic equation. So q = det A and p = -tr A.

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

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Asked about 110 times in JEE Main since 2020. This question is from 2022.