Special Matrices

a) If A and B are idempotent, is AB idempotent? Recall that A is idempotent if A2 = A Prove or give an example where it fails to be true
b) If A is idempotent what is det(A)? Explain
c) Prove that if A is symmetric (AT = A) then A’ is symmetric. i) Prove using matrix algebra ii) Prove using the definition of matrix multiplication d) if A and B are symmetric, is AB symmetric? Prove or give an example where it fails to be true.
e) If A and B are symmetric is A+B symmetric? Prove or give an example where it fails to be true.
f) Say whether the following product AB is singular, non-singular, or depends on the matrices. Explain why. i) A and B are non-singular i) A and B are singular ii) A is singular and B is non-singular
g) If A and B are non-singular, is A+B non singular? Prove or give an example where it fails to be true
h) Is there a relationship between det(A) and det(AT)? Explain
i) If det(A) #0 find det(A-). Justify your answer
j) Is there a relationship between det(A) and det(Ar)? Explain
k) Prove det(I) = 1, I is any n x n identity matrix