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Let A be a square matrix such that AA T = I . Then 1 2 A A + A T 2 + A - A T 2 is equal to

Options

AA 2 + I
BA 3 + I
CA 2 + A T
DA 3 + A T

Correct answer

D. A 3 + A T

Step-by-step solution

Given: AA T = 1 ⇒ A T = A - 1 Now, solving ⇒ 1 2 A A + A T 2 + A - A T 2 = 1 2 A × 2 A 2 + 2 A T 2 + 2 AA T - 2 AA T ⇒ 1 2 A A + A T 2 + A - A T 2 = 1 2 A × 2 A 2 + A T 2 ⇒ 1 2 A A + A T 2 + A - A T 2 = A 3 + A A T 2 Now using, AA T = 1 , ⇒ 1 2 A A + A T 2 + A - A T 2 = A 3 + A T

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