Let A be a matrix such that A\'A = I. These are known as orthogonal matrices Sho

Question

Let A be a matrix such that A'A = I. These are known as orthogonal matrices Show that the product of 2 orthogonal matrices is orthogonal Prove that det A = plusminus 1 show that A'(A-Is) = - (A-I3)' Use part c) to prove that if det A = 1. then det (A-I3) = 0

Explanation / Answer

a)Say A and B are two orthogonal matrices. Therefore, from the orthogonal property: A A' = I and BB' = I where A' is the transpose of A and B' is transpose matrix of B. To prove: AB is an orthogonal matrix. => (AB)(AB)' = I L.H.S. =(AB)(AB)' =ABB'A' (Since, (AB)' = B'A') =AIA' (Since, BB' = I) =AA' Lhs=Rhs b) the definition of an inverse matrix is: A*A^(-1) = I "I" will be the identity matrix. Now you can substitute in A^(T) for A^(-1) since A^(T) = A^(-1): A*A^(T) = I Take the determinant of both sides: det(A*A^(T)) = det(I) Remember that the det(I) is always 1: det(A*A^(T)) = 1 Then you should also remember a theorem det(A*B) = det(A) * det(B): det(A) * det(A^(T)) = 1 And finally there's one more theorem you need to know: that det(A) = det(A^T): det(A) * det(A) = 1 Simplify: (det(A))² = 1 And you get: det(A) = ±1 c) this is simply done by substituting the lhs with a known matrix hope it helps

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