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Question

# Following questions contains statements given in two columns, which have to be matched. The statements in Column−I are labelled as A, B, C and D while the statements in Column−II are labelled as 1,2,3 and 4. Any given statement in Column−I can have correct matching with ONE statement in Column−II.

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Solution

## A) Let A=⎡⎢⎣2−2−4−1341−2−3⎤⎥⎦Now, A2=⎡⎢⎣2−2−4−1341−2−3⎤⎥⎦⎡⎢⎣2−2−4−1341−2−3⎤⎥⎦⇒A2=⎡⎢⎣2−2−4−1341−2−3⎤⎥⎦⇒A2=AHence,A is an idempotent matrix.B) Let A=⎡⎢⎣−5−8035012−1⎤⎥⎦A2=⎡⎢⎣−5−8035012−1⎤⎥⎦⎡⎢⎣−5−8035012−1⎤⎥⎦⇒A2=⎡⎢⎣100010001⎤⎥⎦⇒A2=IHence, A is involutary matrix.C) Let A=13⎡⎢⎣1−22−212−2−2−1⎤⎥⎦⇒AT=13⎡⎢⎣1−2−2−21−222−1⎤⎥⎦Now, ATA=19⎡⎢⎣1−2−2−21−222−1⎤⎥⎦⎡⎢⎣1−22−212−2−2−1⎤⎥⎦=19⎡⎢⎣900090009⎤⎥⎦⇒ATA=IHence, A is orthogonal matrixD) Let A=⎡⎢⎣113526−2−1−3⎤⎥⎦A2=⎡⎢⎣113526−2−1−3⎤⎥⎦⎡⎢⎣113526−2−1−3⎤⎥⎦⇒A2=⎡⎢⎣000000000⎤⎥⎦⇒A2=O where O is zero matrix.Hence, A is a nilpotent matrix of order 2.

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