Let A = [ 1, 2; - 1, 4 ]. If A⁻¹ = αI + βA, α, β ∈ R, I is a 2 × 2 identity matrix then 4(α - β) is equal to :

Solution

Maths sir · 4 steps

Cayley–Hamilton turns A⁻¹ into a combination of I and A, so read α and β straight off.

1. Trace and determinant. For a 2 × 2 matrix, A² - (tr A)A + (det A)I = O. Here tr A = 1 + 4 = 5 and det A = 4 - (-2) = 6.

   A² - 5A + 6I = O

2. Multiply by inverse. Since det A ≠ 0, multiply through by A⁻¹ to get A - 5I + 6A⁻¹ = O. So A⁻¹ = 5/6I - 1/6A, which gives α = 5/6 and β = -1/6.
3. Quick check. 1/6(5I - A) = 1/6(4, -2; 1, 1), which is exactly 1/(det A) adj A.
4. So. α - β = 5/6 + 1/6 = 1, and 4(α - β) = 4, option (d).

The idea to keep. Whenever A⁻¹ is asked as α I + β A for a 2 × 2 matrix, use A² - (trA)A + (det A)I = O and divide by A. Watch the signs.

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