Q11- T
not proposition because they are neither true nor false
2- F
p
q
Pq
T
T
T
T
F
F
F
T
T
F
F
T
3- T
p
P q
q
~p
~q
~p V ~q
~(~p V~q)
T
T
F
F
F
T
T
T
F
F
T
T
F
F
F
T
T
F
T
F
F
F
F
T
T
T
F
F
~(~p V ~q)= p q
4- F
p
~p
P V~p
T
F
T
T
F
T
F
T
T
F
T
T
p
q
T
T

h}
l U D
The given matrix is [I i [1
U U D
From the matrix,
{i} For all non—zero rows, the ﬁrst non—zero number in the row is a ‘l. [which is
Called a leading 1}
{ii} The leading ‘l in the lower row occurs further to the right than the leading
‘l in the h

alt-1P ‘I- LII LI
3 3 3
U 1 2 2 1 [1 *‘rWe added — 2 times the seeond row to the first
_§ 3 7
U U 1 _ 4 2 3
d d d
1 U U 3 2 _-='1
? d d 2
U 1 U 2 1 2 {—WE added — times the third row to the seednd
s ‘ s s 3
0 CI 1 _4 2 3
d d d
and— gtimes the third row to

l U D
The matrix is II] I D
U l] 1
The objective is to verify.r whether the matrix is in row echelon form, reduced row echelon form, both or
neither.
From the matrix,
{i} For all non—zero rows, the ﬁrst non—zero number in the row is a ‘l. [which is
Called

Inversion Algorithm: To ﬁnd the inverse of an invertible matrix 14,
ﬁnd a sequence of elementary rovv operation that A to the identity and
then perform that same sequence of operation on II1 to obtain A'l
Z'mvide feedback [{1}
Eteplofﬁ
Let
3 4 —1
14:1 CI

‘JJ'
The given matrix is
ODD
l D
U 1
D 0
First interchange '1 and 2 columns. Then interchange 2 and 3 celumns.
The matrix
DDS
EDD
1 El 1 U
U l bec emes [I [
U U '3 ﬂ
Frem the matrix,
[i] For all nen—zere rows, the ﬁrst nen—zere number in the row is a 1. [

Consider the following equations
[3} x. + 5.1: —~.|'|§.1'3 =1 Eh} I, +3.!cr2 +.1r.vJl =2
[Ci x. = —?’x2 +3chJ lid} x." +x: +3.13 = 5
1
{E} I.“ —2xz +x3 = 4 if} ax. —J§xz +§x3 = ﬂ”
It is need to determine which are linear equations in xhxz and .:l-:Jl amon

Uonslderthe equation
x. +5.1:2 “Ex; =1
It is need to determine whether the equation is linear or not in 1:.er and x]-
Observe that the equation ,1," + 5.1: —‘,|l§x} :1 does not involve any.r products or roots of the variables
xpxz and x3 _
It is in the fo

1 a
Considerthe equation
x. + 3.1: +3.12:JI = 2
It is need to determine whether the equation is linear or not in rpxz and 1‘].
Observe that the equation x. + 3.1:z +xlx3 = 2 involves products ofthe variables x: and x3.
Comparing the equation x. + 3.7: +JL

The given statement
“Multiplying a linear equation through by zero is an acceptable elementary row
operation” is False.
Provide feedback {El}
Step 2 of5
Reason: Note that
{alelementary row operations are the algebraic operation that are performed on the l

[Cl
Consider the equation
x. =—'.sz +3~JrJ
It is need to determine whether the equation is linear or not in In)": and I]-
On rewriting the given equation, we have
x. +Tx2+3x1 =ﬂ
Observe that the equation x. + 'Fx, +3.151 = ﬂ does not involve any products

The given statement
“A single linear equation with two or more unknowns must always have infinitely many
solutions” is True.
Provide feedback {El}
Step 2 of5
Reason: Note that
(ajljl'i linear equation in the variables X and y is ofthe form
ax+hy =c (on!)

We ean eliminate xfrom the seeond equation by adding —2 times the first equation to the
seeond. This yields the simplified system.
x—y=5
3,}? = —9
From the seeond equation we obtain 3: = —3 and on substituting this value in the first
equation we obtain it

bIEP 1 0T '3
1 1 _
SE _
Letxlzl l i
5 5 111
1 4 1
5—5 E
To find rel—1, we need to reduee A to the identity matrix by row operations and
simultaneously apply these operations to I. For this we will adjoin the identity matrix to
the right side ofJ-"i, there

We need to find the inverse ofthe given matrix hv using the method ofinversion
algorithm
We reduce the given matrix to the identitv matrix lov rovv operations and simultaneouslv
applv these operations to Ito produce the inverse of the given matrix.
Provid

Q1:
1-F
2-T
3-F
4-T
5-F
Q2:
1-B
2-A
3-D
4-C
5-B
Q3If p= a/b and q = c/d be two rational numbers, where a, b, c, d are integers.
Now add them:
p+q
a/b + c/d
(ad)/(bd)+(bc)/(bd)
(ad+bc)/(bd)
So p + q = (ad+bc)/(bd), which is a rational number (the numerator

CH1_MATH
Methods of Elimination
Step 1: Locate the leftmost row that does not contain only zeros.
Step 2: Switch the top row with the closest row that does not start with a zero.
Step 4: Fix the second row through the last row so that the leftmost column

The given statement
“The linear system
x— y=3
2x—2y=k
lCannot have a unique solution, regardless ofthe value of fit?” is True.
Provide feedback {El}
Step 2 of 3
Reason:
I case: We can eliminate it from the second equation by adding —2 times the first
equa

Solution:
The given statement
“Ifﬂ and. B are square matrices ofthe same order, thenrrfrlﬁ’) =ﬁ£ﬂj£r[3)”
is-.
Provide feedback {El}
Step 2 of5
Reason:
No simple relationship exists among? [21:] ,3?" [3) . In particular, we emphasize that
3.?" (AB) will us

inversion Algorithm: To ﬁnd the inverse of an invertible matrix A,
ﬁnd a sequence of elementary rovv operation that A to the identity and
then perform that same sequence of operation on In to obtain A'l
Provide feedback {El}
Step 2 of3
[1 —3‘—1 a]
3 _2 U

x—yz
2x+y = 6
1 —1 1
[2 1 6:1
Add times the first equation to the
seeond to obtain
x—yzl
3}: = 4
1—11
034
Multiply the seeondrow by g to obtain
1 —1 1
CI 1 4—
3
Provide fesdbaeI-s: {El}
Step 5 of5
x—y=1
_ 4
y 3
Adding seeond equation to the first to obtai

The given statement
2 —l 4
0 CI —l
“The linear system with corresponding augmented matrix |: :|is consistent’ is
Provide feedback {El}
Step 2 of 2
Reason:
A linear system is consistent if it has atleast one solution and inconsistent if it
has not solution

Ministry of Higher Education
Kingdom of Saudi Arabia
CSTS
SEU, KSA
Linear Algebra (Math 251)
Level IV, Assignment 3
(Fall, 2016)
1. State whether the following statements are true or false:
[6]
(a) If 2, 3 and 4 are eigen values of a matrix A, then det(A)