SHOW YOUR WORK!!

SOLVE THESE SYSTEMS OF LINEAR EQUATIONS IF POSSIBLE:

A/

1)	2)
3)

(15)

B/

1) Find det (A) 2) Does A – 1 exist? Why or why not?		3) Solve x + 2y – z = 0 2x + 3y + 5z = 0 – x – 3y + 8z = 0

(9)

C/

1) Is A invertible? 2) Find A – 1 if it exists.		4) Use your answer from 2) to solve 3x + y + 4z = 7 4x + y + 6z = – 3 x + z = 2
3) How many solutions are there for
3x + y + 4z = 0 4x + y + 6z = 0 – x + z = 0

(11)

D/

1) Prove that if A, B, and C are invertible matrices of the same size,
then ABC is invertible and (ABC) ^{– 1} = C ^–1B ^–1A^–1

(5)

2) Show that is its own inverse.

(3)

3) Find inverses for each of these elementary matrices.

(3)

4) Find elementary matrices E₁ and E₂ such that E₁A = E₂B.

(4)

TOTAL (50)

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