1. Determine whether the following statements are true or false. If your answer is "true", then prove your answer; if your answer is “false", then give a counterexample that shows that the statement...

Extracted text: 1. Determine whether the following statements are true or false. If your answer is "true", then prove your answer; if your answer is “false", then give a counterexample that shows that the statement is false. (a) If A is an orthogonal matrix, then det(A) = ±1. (Hint: What is A-1?) (b) If det(A) = ±1, then A is an orthogonal matrix. Suppose T: R" → R" is a linear transformation. If, for any pair of orthogonal (c) vectors v and w in R", the vectors T(v) and T(w) are also orthogonal vectors in R", then T is an orthogonal transformation.