Matrices, Math 220, Spring 2006 Instructor: Vitaliy Gyrya

Final, Spring 2003:
Q #17: Why only (c) is the correct answer, when for all matrices columns are orthogonal?
This is a common misconseption resulting from the name of orthogonal matrix. By definition, Matrix is orthogonal when its inverse is the same as transpose (see definition p. 391). This means that columns of the matrix are not only orthogonal, but also normal.

Q: #24 on the Fall Exam and #23 on the Spring Exam are very similar questions just with different numbers. I have no idea how to do either of them. Are they solved the same way?
# 23 is the diagonalization question. The matrix P would consist of eigenvectors of A. which are obviously [1,0]^t and something else. Hence you have to check only (a) and (c). After checking the second column vector you would realize that only (a) has it.

Final, Fall 2003:
Q: I am still confused on #24 on the Fall test.
If you were just to check orthogonality of matricies you would eliminate
"a" - because column-vectors are not normal, and
"d" - because they are not orthogonal.
So than it is just a choice between "b" and "c". It is easy to check that columns of "b" are eigenvectors and those of "c" (after some thinking) are not.