Exercise 9.1. Let R = M2(R), the ring of 2-by-2 matrices with entries in R. (1) Show that a matrix A E R is a zero divisor if and only if det(A) = 0. (2) Show that every left zero divisor of R is also a right zero divisor of R. (This actually holds for R = Mn(R).)

Let A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and nullity of AB. b Show that matrices A and B must be identical.

Consider the matrices below. X=[1201],Y=[1032],Z=[3412],W=[3241] Find scalars a,b, and c such that W=aX+bY+cZ. Show that there do not exist scalars a and b such that Z=aX+bY. Show that if aX+bY+cZ=0, then a=b=c=0.

Prove that if A is similar to B and A is diagonalizable, then B is diagonalizable.