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(MATH113)113midterm2000S.pdf
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Math 113, L2, Midterm Eam, Spring 2000

1. Answer the following questions. Provide a counter example for each false question (3

points for each true question and 5 points for each false question).
m
~

(1). The equation A~x has a solution for any b 2 R if A is an m . n matrix with m . n

and does not contain a zero row.
True ( ). False ( ).
~
(2) The equation A~x . 0 has only trivial solution if A is an m . n matrix with m . n

and does not contain a zero row.
True ( ). False ( ).
(3) If the sets f~v; ; :::; ~v; ~vg and f~v; :::; ~v; ~vg are linearly indep endent, then the

1p.2p.11p.2p
set f~v; :::; ~v; ~v; ~vg is linearly indep endent.
1p.2p.1p
True ( ). False ( ).
(4) If the set f~v; :::; ~v; ~v; ~vg is linearly indep endent, then the subset f~v; :::; ~vg

1p.2p.1p1p.2
is also linearly indep endent.
True ( ). False ( ).
(5)
Since 3a . 3b implies a . b in numbers, we have that the matrix pro duct AB . AC


(6)
The matrix product AB . 0 implies A . 0 or B . 0.



implies B . C if A does not contain a zero row.
True ( ). False ( ).
True ( ). False ( ).

nm

(7) If T is a linear transformation from R to R with the standard matrix A, then T is

mm
onto R if and only if the columns of A span R .
True ( ). False ( ).
(8)
If an n . n matrix contains a zero row, then jAj . 0.

True ( ). False ( ).

(9)
For two n . n matrices A and B, jABj . 0 implies jAj . 0 or jBj . 0.

True ( ). False ( ).

(10)
What are the inverses of the following matrices (1 point for each)


2 32 32 3
150 100 001
4 54 54 5
A. 010 ;B. 020 ;C. 010 :

001 001 100
2. Find the general solution in vector form of the following system of linear equations

(20pts)
8
x.3x+2x+x. .9
.
1 234

.2x+6x.3x. 7

12 4

:
4x.12x.4x+7x. .3
1 234

3. Find the inverse of the matrix by row operations (20pts)

23
1 3 .1
45
A. 012

.1 0

8
4. Compute the following determinant (20pts)
..
..
.3.2 1.4
..
..

1 3 0.3
..
..
.3 4.2 8
..
..
3.4 0 5