Section 5.2: # 4, 7, 12, 13, 16, 21. Section 5.3: # 3, 9, 15, 19, 20. Section 5.

Section 5.2: # 4, 7, 12, 13, 16, 21.
Section 5.3: # 3, 9, 15, 19, 20.
Section 5.4: # 5, 6, 15, 16.
See attached file for questions MAKE SURE TO SOLVE IN STEPS
Hints:
Section 5.3:
#19. To obtain QR factorization, you need to apply Gram-Schmidt process for column vectors of A first. What happens if A is an orthogonal matrix?
#20. Note that det(QR) = det(Q)det(R).
Section 5.4:
#15. A matrix is orthogonally diagonalizable if and only if it is symmetric. So, you can prove that AB is symmetric.
#16. Orthogonally diagonalize the matrix A first. Note that a diagonal matrix D has all nonnegative entries of its diagonal if and only if D=M^2 for some diagonal matrix M.