Question

Let A = `((1, -1, 0),(0, 1, -1),(0, 0, 1))` and B = 7A²⁰ – 20A⁷ + 2I, where I is an identity matrix of order 3 × 3. If B = [b_ij], then b₁₃ is equal to ______.

विकल्प

• 910

• 911

• 912

• 913

Answer 1

Let A = `((1, -1, 0),(0, 1, -1),(0, 0, 1))` and B = 7A²⁰ – 20A⁷ + 2I, where I is an identity matrix of order 3 × 3. If B = [b_ij], then b₁₃ is equal to 910.

Explanation:

Let A = I + C

Where C = `|(0, -1, 0),(0, 0, -1),(0, 0, 0)|`

C² = `|(0, 0, 1),(0, 0, 0),(0, 0, 0)|`

C² = `|(0, 0, 0),(0, 0, 0),(0, 0, 0)|`

So, A² = (I + C)² = I + 2C + C²

A³ = A².A = I + 3C + 3C²

A⁴ = I + 4C + 6C²

A⁵ = I + 5C + 10C²

So, Aⁿ = `"I" + "nC" + ("n"("n" - 1))/2"C"^2`

A²⁰ = I + 20C + 190C²

∴ A⁷ = I + 7C + 21C²

∴ B = 7A²⁰ – 20A⁷ + 2I

B = 7(I + 20C + 190C²) – 20(I + 7C + 21C²) + 2I

∴ B = –11I + 910C²

= `|(-11, 0, 910),(0, -11, 0),(0, 0, -11)|` 

∴ b₁₃ = 910