Question

Let M = `[(0, -α),(α, 0)]`, where α is a non-zero real number an N = `sum_(k = 1)^49`M^2k. If (I – M²)N = –2I, then the positive integral value of α is ______.

Options

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Answer 1

Let M = `[(0, -α),(α, 0)]`, where α is a non-zero real number an N = `sum_(k = 1)^49`M^2k. If (I – M²)N = –2I, then the positive integral value of α is 1.

Explanation:

Given, M = `[(0, -α),(α, 0)]`

and N = `sum_(k = 1)^49`M^2k

Now, M² = `[(0, -α),(α, 0)][(0, -α),(α, 0)]`

= `[(-α^2, 0),(0, -α^2)]`

= `-α^2[(1, 0),(0, 1)]`

= –α²I

Now, N = `sum_(k = 1)^49`M^2k

⇒ N = `sum_(k = 1)(-α^2"I")^"k"`

⇒ N = `sum_(k = 1)^49(-α^2)^"k" "I"`

⇒ N = `(-α^2){((-α^2)^49 - 1)/(-α^2 - 1)}"I"`

⇒ N = `(-α^2(1 + α^98)"I")/(1 + α^2)`

Now, I – M² = I + α²I = (1 + α²)I

∵ (I – M)²N = –2I

⇒ `(1 + α^2)((-α^2(1 + α^98))/(1 + α^2))"I"` = –2I

⇒ α²(1 + α⁹⁸) = 2

⇒ α = 1