98 CHAPTER 4. MATRICES

30. Let

A =

(2 11 2

)Find A−1 if possible. If A−1 does not exist, determine why. Do this in Q2 and in Z2

3.

31. If you have any system of equations Ax= b, let ker(A)≡ {x : Ax= 0} . Show thatall solutions of the system Ax= b are in ker(A)+yp where Ayp = b. This meansthat every solution of this last equation is of the form yp +z where Az = 0.

32. Write the solution set of the following system as the span of vectors and find a basisfor the solution space of the following system. 1 −1 2

1 −2 13 −4 5

 x

yz

=

 000

 .

33. Using Problem 32 find the general solution to the following linear system. 1 −1 21 −2 13 −4 5

 x

yz

=

 124

 .

34. Write the solution set of the following system as the span of vectors and find a basisfor the solution space of the following system. 0 −1 2

1 −2 11 −4 5

 x

yz

=

 000

 .

35. Using Problem 34 find the general solution to the following linear system. 0 −1 21 −2 11 −4 5

 x

yz

=

 1−11

 .

36. Write the solution set of the following system as the span of vectors and find a basisfor the solution space of the following system. 1 −1 2

1 −2 03 −4 4

 x

yz

=

 000

 .

37. Using Problem 36 find the general solution to the following linear system. 1 −1 21 −2 03 −4 4

 x

yz

=

 124

 .