=========================preview======================
(math111)[2008](f)MATH111Q2B~PPSpider^_10428.pdf
Back to MATH111 Login to download
======================================================
MATH111 Quiz 2A Solutions
Page 1/4
Oct 11, 2008 by Daniel Zheng
..................................................40 mins allowed..................................................

Name: Student ID: Score:
..
.
.012 .
...
.
I. [6] Find the matrix A whose inverse is A.1 = 1 0 3 , and compute det A.
4 .38
0 1 2100
..
.
Solutions. [A.1 I]= 103010
4 .38001
..
.
100 .9/27 .3/2
010 .24 .1 =[IA].
001 3/2 .21/2

.
1 .
1
30
1/ det A = det A.1 = . +2 = .2, hence det A = .1/2.
48 4 .3

..
.
MATH111 Quiz 2A Solutions Page 2/4
II. Determine whether the following statements are true or not. If true, justify your answer; if false, give a counterexample. A, B are all SQUARE matrices.
1.
[2] det(.A)= . det A.

2.
[2] If AB is invertible, so is B.


Solutions.
1.
False. Consider A = I2, det(.I2) = 1 = det I2.

2.
True. Since AB is invertible, there exists a matrix W such that W AB = I. In other words, there exists a matrix C = WA such that CB = I. By IMT, B is invertible.


MATH111 Quiz 2A Solutions Page 3/4
III. Let A be an n n matrix.
1.
[3] Suppose Ak = 0 for some k> 1. Find the inverse for I . A if exists.

2.
[3] Suppose A2 . 2A + I = 0. Show that A3 =3A . 2I. What can you conclude for Ak (k> 3)? Justify your conclusion.


Solutions.
1.
(I . A).1 = I + A + + Ak.1 .

2.
Ak = kA . (k . 1)I. Prove by Mathematical Induction.


MATH111 Quiz 2A Solutions Page 4/4
IV. [4] For a triangle with vertices (0, 0), (x1,y1), (x2,y2), its area is given by 1 x1 y1
det .
2
x2 y2 Now in general, for a triangle with vertices (x1,y1), (x2,y2), (x3,y3), show that
..
.
x1 y1 1
.
1
.. det1area= xy.22
2 x3 y3 1
..
.
..
.
Proof. Notethatareaisinvariantundertranslation.Inotherwords,thetriangle underdiscussionhasthe ofthetriangleobtainedbysubtracting sameareaa vector,say(),fromthethreevectorscorrespondingtoitsvertices.Thenewx,x12....trianglehasvertices(00)()().Applytheknownxx,yyxx,yy,,,21213131
formula, we obtain its area
1 2 .
. det . . x2 . x1 x3 . x1 y2 . y1 y3 . y1 . . .
. .
On the other hand,
1 2 .
det . .
. x1 x2 x3 y1 y2 y3 1 1 1 . .
. .
= 1 2 .
det . .
. x1 x2 . x1 x3 . x1 y1 y2 . y1 y3 . y1 1 0 0 . .
. .

= 1 2 .
.
det . . x2 . x1 x3 . x1 y2 . y1 y3 . y1 . . .
.
.
Therefore, proved.