If A = [ 2, 3; 0, - 1 ], then the value of det(A⁴) + det(A¹⁰ - (Adj(2A))¹⁰) is equal to _____________.

Solution

Maths sir · 5 steps

Do not expand the tenth powers: both matrices are upper triangular, so only the diagonal matters.

1. First term. det A = 2 × (-1) - 3 × 0 = -2. So det(A⁴) = (det A)⁴ = (-2)⁴ = 16.
2. Find adj(2A). 2A = (4, 6; 0, -2). For a 2 × 2 matrix, swap the diagonal entries and change the sign of the other two. This gives adj(2A) = (-2, -6; 0, 4).
3. Diagonal of powers. Both matrices are upper triangular, so their powers stay upper triangular, and each diagonal entry is raised to the power. A¹⁰ has diagonal 2¹⁰ = 1024 and (-1)¹⁰ = 1. (adj(2A))¹⁰ has diagonal (-2)¹⁰ = 1024 and 4¹⁰.

   A¹⁰ - (adj(2A))¹⁰ = (1024 - 1024, ∗; 0, 1 - 4¹⁰) = (0, ∗; 0, 1 - 4¹⁰)

4. Second term. The first column of this difference is all zeros, so its determinant is 0. The top-right entry ∗ does not matter.
5. So. The value is 16 + 0 = 16.

The idea to keep. Powers of an upper triangular matrix stay upper triangular, with each diagonal entry raised to that power. And det(P - Q) ≠ det P - det Q, so subtract the matrices first.

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