Answers to quiz 3

1. Matrix M is orthogonal if MT M = I (Eq. 7.9)

2. If matrices A and B are symmetric, their commutator [A,B] = AB−BA is antisymmetric (Sect. 9, Problem 18).
Proof: [A,B]T = (AB−BA)T = (AB)T−(BA)T = BTAT−ATBT = BA−AB = −[A,B]

3. A is orthogonal to A×B by definition of the vector product. B|A|+A|B| and A|B|−B|A| are orthogonal since their scalar product is 0 (Sect. 4, Problem 21).

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