Question

If A = `[(1/sqrt(5), 2/sqrt(5)),((-2)/sqrt(5), 1/sqrt(5))]`, B = `[(1, 0),(i, 1)]`, i = `sqrt(-1)` and Q = A^TBA, then the inverse of the matrix A. Q²⁰²¹ A^T is equal to ______.

Options

• `[(1/sqrt(5), -2021),(2021, 1/sqrt(5))]`

• `[(1, 0),(2021i, 1)]`

• `[(1, 0),(-2021i, 1)]`

• `[(1, -2021i),(0, 1)]`

Answer 1

If A = `[(1/sqrt(5), 2/sqrt(5)),((-2)/sqrt(5), 1/sqrt(5))]`, B = `[(1, 0),(i, 1)]`, i = `sqrt(-1)` and Q = A^TBA, then the inverse of the matrix A. Q²⁰²¹ A^T is equal to `underlinebb([(1, 0),(-2021i, 1)])`.

Explanation:

Given A = `[(1/sqrt(5), 2/sqrt(5)),((-2)/sqrt(5), 1/sqrt(5))]`

Now, AA^T = `[(1/sqrt(5), 2/sqrt(5)),((-2)/sqrt(5), 1/sqrt(5))][(1/sqrt(5), (-2)/sqrt(5)), (2/sqrt(5), 1/sqrt(5))]` 

= `[(1, 0),(0, 1)]` = I  ...(i)

Given Q = A^TBA

So, Q² = (A^TBA)(A^TBA) = A^TB²A

⇒ Q³ = A^TB³A

⇒ Q²⁰²¹ = A^TB²⁰²¹A

Now, let P = AQ²⁰²¹A^T

⇒ P = A(A^TB²⁰²¹A)A^T

Since AA^T = I   ...[From (i)]

⇒ P = B²⁰²¹

Now, B² = `[(1, 0),(i, 1)][(1, 0),(i, 1)] = [(1, 0),(2i, 1)]`

B³ = `[(1, 0),(2i, 1)][(1, 0),(i, 1)] = [(1, 0),(3i, 1)]`

So, B²⁰²¹ = `[(1, 0),(2021i, 1)]`

Inverse of P = (P^–1) = (B²⁰²¹)^–1 = `[(1, 0),(-2021i, 1)]`