Q31[+] Let R be a REF of Amn . Let A1 be obtained by deleting
the last k columns of A and R1 be obtained by deleting the last k columns
of R. Is R1 a REF of A1 ?
A) Yes.
B) No.
Answer (Write A or B .)
Previous
Next
Submit
Q31[+] Let R be a REF of Amn . Le

Q8[+] Which of the following can NOT be [T1,2 (2)]?
A)
"
C)
"
1
0
0
2
1
0
0
0
1
1
0
0
0
1
0
2
0
1
1
0
B) 0
0
#
#
2
1
0
0
0
0
1
0
h
i
1 2
D) 0 1
Answer (Type capital A or B or C or D)
Previous
Next
0
0
0
1
Submit
Q8[+] Which of the following can NOT be [T1

Q15[+] Consider all matrices with complex entries. Define a relation
between two matrices as A is related to B if A can be reduced to B
by a finitely many row operations. Which of the following statement is
correct?
A) This is NOT a reflexive relation.
B)

Q14[+] From now onwards, we shall use Ti (c), Ti,j , Ti,j (c) to denote the elementary row operations and the elementary matrices. Suppose A can be reduced to B by applying the elementary row operations
T1 , T2 , . . . , Tk , in that order (means, first T

Q18[+] Which of the following statement is NOT correct?
A) If A is row equivalent to B, then both can have the same REF.
B) If A is row equivalent to B, then both can have different REF.
C) If A and B can be reduced to the same REF, then they are row
equi

Q19 How many matrices A34 in REF are there with entries 0 or 1
with exactly two pivots?
Answer (Write the number.)
Previous
Next
Submit
Q19 How many matrices A34 in REF are there with entries 0 or 1
with exactly two pivots?
Answer (Write the number.)
Yes.

Q20[+] The augmented coefficient matrix (ACM) of a system of linear
equations Ax = b is the matrix [A | b].
If
"
1
0
0
2
0
0
0
1
#
is the ACM of a system (here means some number),
then the system
A) has infinitely many solutions.
B) has no solutions.
C) h

Q23[+] Fill in the blanks using the options given below to write a
proof of row equivalent systems have the same solution set.
Let [A | b] and [C | d] be row equivalent. Then [C | d] = Tk T1 [A | b] for
T1 , . . . , Tk . Notice that T1 Tk is
, as T1 , . .

Q22[+] If
1
0
0
0
0
1
2
0
0
0
2
0
1
0
0
0
1
0
3
0
0
5
5
0
0
is the ACM of a system, then the system
A) has infinitely many solutions.
B) has no solutions.
C) has exactly one solution.
D) has exactly two solutions.
Answer (Write A or B or C or D.)
Previou

Q24[+] Gaussian elimination To solve Ax = b reduce [A | b] to [C | d]
which is in REF and find the solution set S for [C | d]. Since row equivalent
systems have the same
" solution set,#S is the solution set for Ax = b.
Consider the system
x+y+z =3
x + 2y

Q25[+] Consider a system Amn x = b such that [A | b] is in REF.
Variables corresponding to pivotal columns are called basic variables.
Other variables
" are called free#variables. Which of the following is correct
for the ACM
1
0
0
3
0
0
0
2
0
1
2
0
0
2
3

Q26[+] Fill in the blanks. Consider a system Amn x = b such that
[A | b] is in REF. Suppose that A has k pivots and the last column b is
th row of A is a zero row and bk+1
0.
also pivotal. Then
Hence, the
th equation cannot be satisfied by giving any valu

Q7[+] Which of the following can NOT be [T1,2 ]?
A)
"
C)
"
0
1
0
1
0
0
0
0
1
0
1
0
0
0
1
1
0
0
0
1
B) 0
0
#
#
1
0
0
0
0
0
1
0
h
i
0 1
D) 1 0
Answer (Type capital A or B or C or D)
Previous
Next
0
0
0
1
Submit
Q7[+] Which of the following can NOT be [T1,2

Q3[+] Take Amn . Let Ti,j (c) be the operation of adding c times the
jth row to the ith row with. Which of the following is NOT correct (in
general) for A43 ?
A) T1,3 (c)T1,3 (c)A = A
B)T1,3 (c)T1,3 (c)A = T1,3 (2c)A
C) T1,3 (c)T1,4 (c)A = T1,4 (c)T1,3 (c

Q6[+] If Ti is an elementary row operation, then by [Ti ] we denote
the corresponding elementary matrix. Which of the following can NOT
be the size of [T3 (2)]?
A) 8
C) 4
B) 6
D) 2
Answer (Type capital A or B or C or D)
Previous
Next
Submit
Q6[+] If Ti is

Q33[+] Let Amn and Bmn be in REF. Assume that A has k pivots
but last column is not pivotal and B has k pivots in the same columns
as that of A but B has one more pivot in the last column. Are they row
equivalent?
A) Yes
B) No.
Answer (Write A or B .)
Pre

Q48[+] Take A =
"
2
1
3
4
2
6
0
3
1
We must have B:1 = e1 , as
6
3
9
6
6
10
#
7
7
7
. Let us find B = RREF (A).
.
As A:2 = 2A:1 , we must have B:2 =
.
As A:3 is not a scalar multiple of A:1 , we must have B:3 =
As A:4 = A:1 + A:2 , we must have B:4 =
As A

DEPARTMENT OF MATHEMATICS, IIT Guwahati
MA101: Mathematics I, July - November 2014
1. Let A and B be two m n matrices that are in reduced row echelon form. If the systems Ax = 0 and Bx = 0
have the same solution set then show that A = B.
Solution: We will

MA 101 (Mathematics I)
Practice Problem Set
1. State TRUE or FALSE giving proper justification for each of the following statements.
(i) If both (xn ) and (yn ) are unbounded sequences in R, then the sequence (xn yn ) cannot be
convergent.
(ii) If both (x

MA 101 (Mathematics I)
Hints/Solutions for Examples/Exercises in Lectures
Sequence
Exercise: Examine whether the sequence (1)n n1 ) is convergent. Also, find the limit if it exists.
Solution: Given > 0, there exists n0 N such that n0 > 1 . For all n n0 ,

MA 101 (Mathematics I)
Solutions for Tutorial Problem Set
1. Let (xn ) be a convergent sequence of positive real numbers such that lim xn < 1. Show that
n
lim xnn = 0.
n
Solution: If ` = lim xn , then 12 (1`) > 0 and so there exists n0 N such that |xn `|

Q51[+] Take A919 and let B = RREF (A). Assume that B:1 , B:3 , B:5 ,
and B:9 are the only pivotal columns in B. Which of the following are
correct?
A) The matrix At has 19 rows.
B) We can first apply some elementary row operations to At to make its
2nd ro

Q54[+] Select the correct statements.
A)
is an equation in two variables. It represents the set
x + y = 3
3t
: t R in R2 which is a straight line.
t
B) The inequation x + y < 3 also represents a subset of R2 .
C) The equation x21 + x22 = 1 represents a

Q57[+] Let S be the line x + y = 3 and T be the line x + y = 0 in
R2 . Which of the following is NOT correct?
3
2
A) S = 0 + T
B) S = 1 + T
1
2
C) S = 2 + T
D) S = 1 + T
Answer (Write like ABD or BC.)
Previous
Next
Submit
Q57[+] Let S be the line

Q59[+] Consider the system Ax = b and let Li denote the set (subset
of Rn ) for the equation given by row i. The system Ax = 0 is called the
associated homogeneous system (AHS) and let Li denote the set (subset
of Rn ) for the equation given by row i. Let

Q2[+] Take A43 . By Ti,j Tl,k A we mean first apply Tl.k on A, then
apply Ti,j on the result. Which of the following is NOT correct (in
general)?
A) T1,3 T1,3 A = A
B)T1,3 T2,4 A = T2,4 T1,3 A
C) T1,3 T1,4 A = T1,4 T1,3 A
D) T1,3 A =
0
0
1
0
0
1
0
0
1
0
0

Q28[+] Take Amn x = 0 such that [A | 0] is in REF. Suppose that
each variable is basic. Then
A) The system has a trivial solution (means each xi = 0), but it may
have more solutions.
B) The system has the trivial solution, and by back substitution it cann

Q34[+] Let Amn and Bmn be in REF. Assume that A and B are
row equivalent. Must they have the same pivotal positions?
A) Yes
B) No.
Answer (Write A or B .)
Previous
Next
Submit
Q34[+] Let Amn and Bmn be in REF. Assume that A and B are
row equivalent. Must

Q35[+] Let Ann be invertible and R be a REF of A. Should R have
n pivots?
A) Yes
B) No.
Answer (Write A or B .)
Previous
Next
Submit
Q35[+] Let Ann be invertible and R be a REF of A. Should R have
n pivots?
A) Yes
B) No.
Answer (Write A or B .)
Correct. N