5. Let A and B be the matrices 3 B = (a) Calculate A" A, simplify as much as possible. (b) Calculate BT B, simplify as much as possible. (c) Calculate (AB) (AB). HINT: If you use parts (a) and (b), together with some of the rules for matrix multiplication, it is not necessary to calculate AB. 6. A real matrix A is called normal if AAT - ATA. (a) Why is a non-square matrix never normal? (b) Find all real values of a such that the matrix 4 - ( 2 1 ) is normal. 7. Let t be a positive real number, and let u, u and w be the vectors U - 2j, V21 2VE 2VE - k , (a) Let 0 be the angle between u and u. Determine cos(0) as a function of t. (b) Find the angle between u and w. Give your answer as a (rational) multiple of 7 in radians. (c) Find a vector that is perpendicular to both u and w. sin($ + 0) - sin() cos(0) # cos(p) sin(0) cos(6 + 0) - cos(@) cos(0) + sin() sin(0) 2 cos(0) cos() - cos(0 $) + cos(0 + ), 2 sin(0) sin () - cos(0 - 6) - cos(0 + (), 2 sin(0) cos() - sin(0 + o) + sin(0 ), 2 cos(0) sin(6) - sin(0 + 4) - sin(0 - )