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(MATH113)[2010](f)quiz~2407^_10452.pdf
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MATH 113 Introduction to Linear Algebra Name: Quiz 1 for T6a
Student ID: Time allowed: 20 minutes
1. Find the inverse of the following matrix:

110 0
01 .10

00 1 .1
000 1


A =

.

2. Suppose that B is an n n matrix satisfying (B2 + In)(B2 . In)= O. Show that B.1 = B3 and simplify B9 .
END
1. Perform row operations on [ A | I4 ]: 110 0

1000 1100
1000
.
.
.
.
...

01 .10

0100
00 1 .1

0010
...


r4+r3
.

...


01 .10
0100
00 1 0
0011
...

000 1

0001 0001
0001
.
.
.
.
1100

1000 1000
1 .1 .1 .1
r3+r2
.



...

0100

0111
0010

0011
...

.r2+r1
.



...

0100
01 11
0010
00 11
...

0001

0001 0001
00 01
Hence the inverse of A is:


.
.
...

1 .1 .1 .1
01 1 1

00 1 1
00 0 1

...

A.1

=
.

2. By direct expansion, we have (B2 + In)(B2 . In)= B4 . In = O. So B B3 = In and hence B.1 = B3
. Then
B9 = B8 B =(B4)2 B = In B = B.

MATH 113 Introduction to Linear Algebra Name: Quiz 1 for T6b
Student ID: Time allowed: 20 minutes
1. Find the inverse of the following matrix:

1 .100
01 .10

00 11
00 01


A =

.

2. Suppose that B is an n n matrix satisfying In + B + B2 = O. Show that B.1 = B2 and simplify B10 .
END
1. Perform row operations on [ A | I4 ]: 1 .100

1000
.
.
.
.


1 .100
100 0
...

01 .10

0100
00 11

0010
...


.r4+r3
.

...


01 .10
010 0
00 10
001 .1
...

00 01

0001 00 01
000 1
.
.
.
.
1 .100

100 0 1000
111 .1
r3+r2
.



...

0 1 00

011 .1
0 0 10

001 .1
...

r2+r1
.



...

0100
011 .1
0010
001 .1
...

0 0 01

000 1 0001
000 1
Hence the inverse of A is:


.
.
...

111 .1
011 .1

001 .1
000 1

...

A.1
=
.


2. Since In + B + B2 = O . B(.In . B)= In, so: B.1
= .In . B = .In . B +(In + B + B2)= B2 . Then B3 = B2 B = B.1 B = In and: B10 =(B3)3 B = In B = B.