Math 369 HW #3
Due at the beginning of class on Friday, September 19
Reading: Sections MISLE, LC, SS.
Problems:
1. Let
A=
−1 2
−4 3
andB =
3/5
4/5
−2/5
−1/5
(a) Show that A and B are inverses of eachother.
(b) Solve the following system of equations by using B appropriately (i.e., not by rowreducing):
−x + 2y =
1
−4x + 3y = −2
*
2. Does the matrix have an inverse? If so, find it. If not, find some vector x which is not the
*
*
zero vector so that A x = 0.
(a)


1
2 −2
−1 5 −3
5 −4 0
(b)


1 0 2
0 3 0
3 0 8
a b
∈ Mat2,2 (R) and define the matrix
3. (a) Let A =
c d
d −b
∗
A =
.
−c a
Show that AA∗ = A∗ A = (ad − bc)12 .
(b) If ad − bc 6= 0, then the matrix
1
B=
A∗ =
ad − bc
d
ad−bc
−c
ad−bc
makes sense. Show that B = A−1 .
4. Suppose B, C ∈ Matn,n (R) are both invertible.
(a) Prove that
(BC)−1 = C −1 B −1 .
1
−b ad−bc
a
ad−bc
(b) Find specific 2 × 2 matrices B and C so that (BC)−1 6= B −1 C −1 .
5. Use Sage to help you solve the following system of equations:
x1
−x1
x1
−x1
2x1
+
−
+
+
+
2x2
2x2
x2
5x2
2x2
+
+
+
+
+
3x3
3x3
x3
3x3
2x3
+
−
+
−
+
4x4
4x4
x4
2x4
3x4
+ 5x5
+ 5x5
+ x5
+ x5
+ 3x5
+ 6x6
− 6x6
+ x6
− x6
+ 3x6
=
=
=
=
=
7
7
2
1
4
(Hint: remember, you can create your own Sage worksheets on the Sage Math Cloud:
http://sagemath.com. Also, clicking the little printer button on a Sage worksheet on the
Sage Math Cloud will generate a PDF which you can then print or email.)
2