Show that the map f(A) = |A| from the set of 2 × 2 matrices to the set of real numbers is not one - one.

Use an example chosen from 22 matrices to show that for nn matrices A and B,ABBA but AB=BA.

Find two 22 matrices such that |A|+|B|=|A+B|.

Let A and B be square matrices of order 4 such that |A|=4 and |B|=2. Find a |BA|, b |B2|, c |2A|, d |(AB)T|, and e |B1|.

Determine if the statement is true or false. If the statement is false, then correct it and make it true. For the product of two matrices to be defined, the number of rows of the first matrix must equal the number of columns of the second matrix.

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.

Can a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a 22 matrix.

Use elementary matrices to find the inverse of A=100010abc, c0.

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.

Give examples of matrices A and B of the same size such that (a) rank(A+B)rank(A) and rank(A+B)rank(B) (b) rank(A+B)=rank(A) and rank(A+B)=rank(B) (c) rank(A+B)rank(A) and rank(A+B)rank(B).