Question

If A = `[(1, 1, 1),(0, 1, 1),(0, 0, 1)]` and M = A + A² + A³ + .... + A²⁰, then the sum of all the elements of the matrix M is equal to ______.

Options

• 2010

• 2020

• 2030

• 2040

Answer 1

If A = `[(1, 1, 1),(0, 1, 1),(0, 0, 1)]` and M = A + A² + A³ + .... + A²⁰, then the sum of all the elements of the matrix M is equal to 2020.

Explanation:

Aⁿ = `[(1, n, (n(n + 1))/2),(0, 1, n),(0, 0, 1)]`

Since, `sum1 = 20, sum_(n = 1)^20n = (20 xx 21)/2` = 210

and `1/2sum_(n = 1)^20n(n + 1) = 1/2 xx (20 xx 21 xx 22)/3` = 1540

∴ M = A + A² + .......... + A²⁰

= `[(sum1, sumn, sum(n(n + 1))/2),(0, sum1, sumn),(0, 0, sum1)]`

= `[(20, 210, 1540),(0, 20, 210),(0, 0, 20)]`

∴ Sum of all the elements of M = 20 + 20 + 20 + 210 + 210 + 1540 = 2020