MTH 261 Portland Community College Solve the Following Systems of Equations Questions

MTH 261 Exam 1 Review1. Solve the following systems of equations given using augmented matrices, converting them
each to reduced-row-echelon forms, and then stating the solution to the system. What are
the ranks of the augmented matrices? What dimension is the solution space of each? What
kind of shape is it?
a.
2x − y + 5z = 10
−x − y − 11z = 19
x + 3y + 17z = −27
b.
x1 − x2 + 2×3 + x4
−x1 + x2 − 2×3 − x4
x1 + x2 + 8×3 + 9×4
−2×1 + x2 − 7×3 − 6×4
=2
= −2
= −18
=6
1
⎤
⎡ ⎤
⎡ ⎤
⎡ ⎤
⎡ ⎤
1
−2
3
4
−1
⎢−1⎥
⎢1⎥
⎢2⎥
⎢0⎥
⎢9⎥
⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
2. Let x = ⎢
⎣ 0 ⎦, y = ⎣1⎦, and z = ⎣−1⎦. If v = ⎣−2⎦ and w = ⎣ 2 ⎦, either write v and w
2
0
1
5
6
as linear combinations of x, y, and z, or show that it cannot be done.
⎡
⎡
⎡ ⎤
⎡ ⎤
⎡ ⎤
⎤
⎡ ⎤
1
−2
0
2
5
⎣
⎣
⎣
⎣
⎦
⎦
⎦
⎦
⎣
3. Let a1 = −1 , a2 = 1 , a3 = −1 , a4 = −1 , and b = −6⎦. Also, let A = [aj ].
−1
2
3
1
16
Show how the equation Ax = b is equivalent to finding the scalars for determining if b is a
linear combination of aj and also equivalent to solving a system of equations. That is, write
down all three interpretations of this problem. Then solve for x for b or show that no
solution exists.
2
⎡
⎤
⎡ ⎤
2
1 −1 −1
1
⎢3
⎥
⎢5⎥
1
1
−2
⎥ and b = ⎢ ⎥. Solve Ax = 0 and write the solutions as a
4. Let A = ⎢
⎣−1 −1 2
⎣2⎦
1⎦
−2 −1 0
2
−2
linear combination of basic solutions. Then solve Ax = b and express every solution as the
sum of a specific solution plus a solution to the associated homogeneous system.

5. What is the determinant and inverse of A =
2 −3
?
−1 1
⎤
1 1 −2
6. Find the inverse of A = ⎣3 −2 3 ⎦ using row reduction. Then use it to solve the system:
3 2 −1
⎡
x + y − 2z = 5
3x − 2y + 3z = −13
3x + 2y − z = 1
3
7. In each case either show that the statement is true or give an example showing it is false.
a. If the a31 entry of A is 5, then the a13
entry of AT is −5.
e. If A and B are both invertible, then
(A−1 B)T is invertible.
b. A and AT have the same main diagonal
for every matrix A.
f. If A is invertible and skew symmetric
(AT = −A), then the same is true of A−1 .
c. If B is symmetric and AT = 3B, then
A = 3B.
g. If A2 is invertible, then A is invertible
d. If A and B are symmetric, then kA + mB
is symmetric for any scalars k and m.
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h. If AB = I, then A and B commute.

T −1
8. Find A if ((I − 2A )
=
2 1
1 1
⎡
⎤
⎡ ⎤
⎡ ⎤


2
3
−6
3
0
3
2
⎣
⎦
⎣
⎦
⎣
−1
−2
9. If T : R → R is a linear transformation, T
=
, and T
=
, find T −5⎦.
6
5
3
1
−4
⎡
⎤

−1 2
1 −2 3 1
. Determine a single matrix which induces the
10. Let A = ⎣ 0 −1⎦ and B =
−1 1 1 2
2
3
same transformation as the composite transformation of TB and then TA . Show a
transformation diagram discussing the domain, co-domain, and the linked domain.
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−1 0
0 1
11. Given A =
and B =
, what transformations are induced by A, B, AB, and
0 1
−1 0
BA? Show an example point on a plane being reflected or rotated by each and indicate how
AB and BA are composite transformations of A and B respectively.
12. Let Q0 : R2 → R2 be reflection in the x-axis, Q−1 : R2 → R2 be reflection in y = −x, and
R−π/2 : R2 → R2 be clockwise rotation by π/2. What are the matrices for the
transformations Q0 , Q−1 , and R−π/2 ? What transformations to you obtain by the composite
of these three transformations? That is, find Q0 ◦ R−π/2 , Q0 ◦ Q−1 , Q−1 ◦ Q0 , Q−1 ◦ R−π/2 ,
R−π/2 ◦ Q0 , and R−π/2 ◦ Q−1 . Do you think this is a contained group of transformations and,
if so, what are all of the transformations which belong in this group?
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