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Choose the correct answer in the following questions If A = [αβy-a] is such that A2 = I, then -

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Question

Choose the correct answer in the following questions

If A = `[(alpha, beta),(y, - a)]` is such that A2 = I, then

Options

  • 1 + α2 + βy = 0

  • 1 – α2 + βy = 0

  • 1 – α2 – βy = 0

  • 1 + α2 – βy = 0

MCQ

Solution

1 – α2 – βy = 0

Explanation:

Given, A = `[(alpha, beta),(y, - alpha)]`

∴ A2 = A, A = `[(alpha, beta),(y, -alpha)][(alpha, beta),(y, -alpha)]`

= `[(alpha^2 + betay, alphabeta - alphabeta),(alphay - alphay, betay + alpha^2)] = [(alpha^2 + betay, 0),(0, betay + alpha^2)]`

Now, A2 = I

⇒ `[(alpha^2 + y, 0),(0, betay + alpha^2)] = [(1, 0),(0, 1)]`

We have equalting the corresponding elements.

`alpha^2 + betay` = 1

⇒ `alpha^2 + betay - 1` = 0

⇒ `1 - alpha^2 - betay` = 0

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