Page 51 - 35Linear Algebra

2.2 Review Problems 51

8. Show that this pair of augmented matrices are row equivalent, assuming
ad − bc 6= 0:
! de−bf !
a b e 1 0
∼ ad−bc
c d f 0 1 af−ce
ad−bc
9. Consider the augmented matrix:

2 −1 3
.
−6 3 1
Give a geometric reason why the associated system of equations has
no solution. (Hint, plot the three vectors given by the columns of this
augmented matrix in the plane.) Given a general augmented matrix

a b e
,
c d f
can you find a condition on the numbers a, b, c and d that corresponds
to the geometric condition you found?

10. A relation ∼ on a set of objects U is an equivalence relation if the
following three properties are satisfied:

• Reflexive: For any x ∈ U, we have x ∼ x.
• Symmetric: For any x, y ∈ U, if x ∼ y then y ∼ x.

• Transitive: For any x, y and z ∈ U, if x ∼ y and y ∼ z then x ∼ z.

Show that row equivalence of matrices is an example of an equivalence
relation.
(For a discussion of equivalence relations, see Homework 0, Problem 4)

Hint

11. Equivalence of augmented matrices does not come from equality of their
solution sets. Rather, we define two matrices to be equivalent if one
can be obtained from the other by elementary row operations.
Find a pair of augmented matrices that are not row equivalent but do
have the same solution set.

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