{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Adv Alegbra HW Solutions 61

Adv Alegbra HW Solutions 61 - 61 If AB = B A then cw dy =...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
61 If AB = B A , then c w + dy = ay + cz ; that is, c (w z ) = y ( a d ) . There are two cases to consider. If a = d , then de fi ne B = 0 1 1 1 . The 2,1 entry of AB is d = a , while the 2,1 entry of B A is a + c . As c = 0, we have AB = B A . If a = d , de fi ne B = 1 1 1 0 . Now the 2,1 entry of AB is c + d , while the 2,1 entry of B A is c + a . Since a = d , these entries are different, and so AB = B A in this case as well. Therefore, A is not in the center of GL ( 2 , R ) . A similar argument holds if b = 0, (This result is generalized to n × n matrices in Corollary 4.86. The proof using linear transformations is much shorter and much simpler.) 2.85 Let ζ = e 2 π i / n be a primitive n th root of unity, and de fi ne A = ζ 0 0 ζ 1 and B = 0 1 1 0 . (i) Prove that A has order n and that B has order 2. Solution. It is clear that A n = I = B 2 . If 1 k < n , then A k = I , for the 1,1 entry of A k is ζ k = 1. (ii) Prove that B AB = A 1 . Solution. One multiplies the matrices. (iii) Prove that the matrices of the form
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}