Question

Let A = `[(2, -1),(0, 2)]`. If B = I – ⁵C₁(adj A) + ⁵C₂(adj A)⁵, –... –⁵C₅(adj A)⁵, then the sum of all elements of the matrix B is ______.

पर्याय

• –5

• –6

• –7

• –8

Answer 1

Let A = `[(2, -1),(0, 2)]`. If B = I – ⁵C₁(adj A) + ⁵C₂(adj A)⁵, –... –⁵C₅(adj A)⁵, then the sum of all elements of the matrix B is –7.

Explanation:

Given A = `[(2, -1),(0, 2)]`

And B = I – ⁵C₁(adj A) + ⁵C₂(adj A)² ... ⁵C₅(adj A)⁵

⇒ B = (I – adj A)⁵

Now, adj A = `[(2, 1),(0, 2)]`

∴ B = `([(1, 0),(0, 1)] - [(2, 1), (0, 2)])^5`

⇒ B = `[(-1, -1),(0, 1)]^5`

⇒ B = `-[(1, 1),(0, 1)]^5`

Now `[(1, 1),(0, 1)]^2 = [(1, 1),(0, 1)][(1, 1),(0, 1)] = [(1, 2),(0, 1)]`

⇒ `[(1, 1),(0, 1)]^3 = [(1, 2),(0, 1)][(1, 1),(0, 1)] = [(1, 3),(0, 1)]`

Sim: Jarly, `[(1, 2),(0, 1)]^5 = [(1, 5),(0, 1)]`

So, B = `-[(1, 5),(0, 1)] = [(-1, -5),(0, -1)]`

∴ Modulus of sum of all elements of matrix B = |–1 –5 – 1| = –7.