Question

Let A = `[(0, -2),(2, 0)]`. If M and N are two matrices given by M = `sum_(k = 1)^10 A^(2k)` and N = `sum_(k = 1)^10 A^(2k - 1)` then MN² is ______.

विकल्प

• a non-identity symmetric matrix

• a skew-symmetric matrix

• neither symmetric nor skew-symmetric matrix

• an identify matrix

Answer 1

Let A = `[(0, -2),(2, 0)]`. If M and N are two matrices given by M = `sum_(k = 1)^10 A^(2k)` and N = `sum_(k = 1)^10 A^(2k - 1)` then MN² is a non-identity symmetric matrix.

Explanation:

Given matrix is

A = `[(0, -2),(2, 0)]`

A² = `[(0, -2),(2, 0)][(0, -2),(2, 0)] = [(-4, 0),(0, -4)]` = –4I

A³ = –4A

A⁴ = (–4I) (–4I) = (–4)²I

A⁵ = (–4)² A, A⁶ = (–4)³I

Take, M = `sum_(k = 1)^10 A^(2k)`

= A² + A⁴ + .... + A²⁰

= [–4 + (–4)² + (–4)³ + ..... + (–4)¹⁰]I

It is GP. with a = –4, r = –4 and n = 10.

S₁₀ = `((-4))/5(2)^10 - 1I`

`\implies` M is symmetric matrix

N = `sum_(k = 1)^10A^(2k - 1)`

= A + A³ + .... + A¹⁹

= A[1 + (–4) + (–4)² + ... + (–4)⁹]

Similarly, N = `A[[1(-4)^10 - 1)/((-4 - 1))]`

= `(A[(2)^20 - 1])/5`

So, N is a skew-symmetric matrix

`\implies` N² is a symmetric matrix

Therefore MN² is a non-identity symmetric matrix.