Begin2.DVI

I10-10. (a) IfAT=AandBT=B, then(AB)T=BTAT=BA=AB

(b) If((AB)T= (AB), thenBTAT=ABwhich impliesBA=AB

I10-12. IfAB=BA, then one must havec= 0andd=a.

I10-13. ShowA^2 =AandA^3 =A, then showB^2 =IandB^3 =BsoAis idempotnt

andBis involutory.

I10-14. ShowA^2 =IandA^3 =Aso thatAis involutory.

I10-15. ShowB=A^2

I10-16. X=A−^1 B

I10-18. (a) -11 (b) 6 (c) -2

I10-19. (a) 8 (b) 5 (c) 4

I10-20. (a) (mij) =





14 − 5 − 8

2 1 − 1

− 3 −1 2





(b) (cij) =





14 − 5 − 8

−2 1 1

−3 1 2





(c) ACT=





1 0 0

0 1 0

0 0 1





I10-21. 6 xyz

I10-22. (a) 0 (b) 0 (c) 36

(d)a 1 a 2 a 3 a 4 (e)a 1 a 2 a 3 a 4 (f) 45 ,000 = (36)(25)(5)(2)(5)

I10-23. (a)Z−^1 =

1

(z 11 z 22 −z 12 z 21 )

[

z 22 −z 12

−z 21 z 11

]

I10-24. |A|= 3, |B|= 1, |A|·|B|= 3, |AB|= 3

I10-25. (x 2 −x 1 )y−(y 2 −y 1 )x=x 2 y 1 −x 1 y 2

Solutions Chapter 10