PrMT - AMS210.01. Practice Midterm Andrei Antonenko 1. A...

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AMS210.01. Practice Midterm Andrei Antonenko 1. A square matrix Q is called orthogonal if QQ > = Q > Q = I . (a) Give an example of an orthogonal matrix. Solution: Identity matrix is orthogonal: since I > = I , we have II > = I > I = II = I . Of course, you can find other orthogonal matrices. (b) A matrix is called inversion matrix if it is obtained from the identity matrix by inter- changing two of its rows. Prove that any inversion matrix is orthogonal. Solution: Let E be an inversion matrix, obtained from I by interchanging rows, say, i and j . Since interchanging rows of I is the same as interchanging columns of I , E > = E . Also, to get I from E we have to interchange the same rows – i and j , and thus E - 1 = E , and thus EE > = EE = EE - 1 = I . (c) Prove that the product of two orthogonal matrices is an orthogonal matrix. Solution: Let A and B be orthogonal matrices. It means that AA > = I and BB > = I . Then (
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PrMT - AMS210.01. Practice Midterm Andrei Antonenko 1. A...

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