Download PTU B.Tech 2020 March CSE-IT 1st Sem MA 1300 Linear Algebra For Engineers Question Paper

Download PTU (I.K. Gujral Punjab Technical University Jalandhar (IKGPTU) ) BE/BTech CSE/IT (Computer Science And Engineering/ Information Technology) 2020 March 1st Sem MA 1300 Linear Algebra For Engineers Previous Question Paper

1 | M-77256 (S1)-2591

Roll No. Total No. of Pages : 03
Total No. of Questions : 09
B.Tech. (Software Engineering) (Sem.?1)
LINEAR ALGEBRA FOR ENGINEERS
Subject Code : MA-1300
M.Code : 77256
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks
each.
2. SECTION - B & C. have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
4. Select atleast TWO questions from SECTION - B & C.

SECTION-A
1. Solve the following :
a) Find the general solution of the linear system whose augmented matrix is
1 3 5 0
0 1 1 1
? ? ? ?
? ?
? ?
? ?
.
b) Reduce the matrix
1 3 5
2 1 4
2 8 2
? ?
? ?
?
? ?
? ?
?
? ?
to row echelon form.
c) Find the inverse of the matrix
1 3
2 4
? ?
? ?
? ?
.
d) Examine whether the transformation T : R
2
? R
2
defined as
| |
?
x x
T
y y
? ? ? ?
?
? ? ? ?
? ? ? ?
is linear or
not?
e) Let A
a b
c d
? ?
?
? ?
? ?
and let k be a scalar. Find a formula that relates det kA to k and det
A.
f) Let a
2
5
1
? ?
? ?
? ?
? ?
? ? ?
? ?
and b
7
4
6
? ? ?
? ?
? ?
? ?
? ?
? ?
. Compute ||a + b||
2
.
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1 | M-77256 (S1)-2591

Roll No. Total No. of Pages : 03
Total No. of Questions : 09
B.Tech. (Software Engineering) (Sem.?1)
LINEAR ALGEBRA FOR ENGINEERS
Subject Code : MA-1300
M.Code : 77256
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks
each.
2. SECTION - B & C. have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
4. Select atleast TWO questions from SECTION - B & C.

SECTION-A
1. Solve the following :
a) Find the general solution of the linear system whose augmented matrix is
1 3 5 0
0 1 1 1
? ? ? ?
? ?
? ?
? ?
.
b) Reduce the matrix
1 3 5
2 1 4
2 8 2
? ?
? ?
?
? ?
? ?
?
? ?
to row echelon form.
c) Find the inverse of the matrix
1 3
2 4
? ?
? ?
? ?
.
d) Examine whether the transformation T : R
2
? R
2
defined as
| |
?
x x
T
y y
? ? ? ?
?
? ? ? ?
? ? ? ?
is linear or
not?
e) Let A
a b
c d
? ?
?
? ?
? ?
and let k be a scalar. Find a formula that relates det kA to k and det
A.
f) Let a
2
5
1
? ?
? ?
? ?
? ?
? ? ?
? ?
and b
7
4
6
? ? ?
? ?
? ?
? ?
? ?
? ?
. Compute ||a + b||
2
.
2 | M-77256 (S1)-2591

g) Show that similar matrices have same eigen values.
h) If ? is an eigen value of A, show that ?
?1
is an eigen value of A
?1
.
i) Check whether the vectors u
12
3
5
? ?
? ?
?
? ?
? ? ?
? ?
and v
2
3
3
? ?
? ?
? ?
? ?
? ?
? ?
are orthogonal or not?
j) The characteristic roots of
8 6 2
6 4
2 4 3
A k
? ? ?
? ?
? ? ?
? ?
? ?
?
? ?
are 0, 3, 15. Find the value of k.

SECTION-B
2. a) Determine if the following system is consistent :
y ? 4z = 8
2x ? 3y + 2z = 1
4x ? 8y + 12z = 1
b) Let u =
1 2 4
4 , 3 and 1
2 7
v w
h
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
, For what value(s) of h is w in the plane spanned
by u and v ?
3. a) Given
1 2 4
0 1 5
2 4 3
A
? ?
? ?
?
? ?
? ?
? ? ?
? ?
and
2
2
9
b
? ? ?
? ?
?
? ?
? ?
? ?
, write the augmented matrix for the linear
system that corresponds to the matrix equation Ax = b. Then solve the system and
write the solution as a vector.
b) Find the inverse of the matrix
0 1 2
1 0 3
4 3 8
? ?
? ?
? ?
? ?
?
? ?
using row transformations.
4. Let T : R
3
? R
3
be a linear transformation defined by
0 x
T y x y
z x y z
? ? ? ?
? ? ? ?
? ?
? ? ? ?
? ? ? ?
? ?
? ? ? ?
. Find the
matrix representation of T w.r.t. the ordered basis B
1
= {(1, 0, 0), (0, 1, 0), (0, 0, 1)} and
B
2
= {(1, 0, 1), (1, 1, 0), (0, 1, 1)}.
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1 | M-77256 (S1)-2591

Roll No. Total No. of Pages : 03
Total No. of Questions : 09
B.Tech. (Software Engineering) (Sem.?1)
LINEAR ALGEBRA FOR ENGINEERS
Subject Code : MA-1300
M.Code : 77256
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks
each.
2. SECTION - B & C. have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
4. Select atleast TWO questions from SECTION - B & C.

SECTION-A
1. Solve the following :
a) Find the general solution of the linear system whose augmented matrix is
1 3 5 0
0 1 1 1
? ? ? ?
? ?
? ?
? ?
.
b) Reduce the matrix
1 3 5
2 1 4
2 8 2
? ?
? ?
?
? ?
? ?
?
? ?
to row echelon form.
c) Find the inverse of the matrix
1 3
2 4
? ?
? ?
? ?
.
d) Examine whether the transformation T : R
2
? R
2
defined as
| |
?
x x
T
y y
? ? ? ?
?
? ? ? ?
? ? ? ?
is linear or
not?
e) Let A
a b
c d
? ?
?
? ?
? ?
and let k be a scalar. Find a formula that relates det kA to k and det
A.
f) Let a
2
5
1
? ?
? ?
? ?
? ?
? ? ?
? ?
and b
7
4
6
? ? ?
? ?
? ?
? ?
? ?
? ?
. Compute ||a + b||
2
.
2 | M-77256 (S1)-2591

g) Show that similar matrices have same eigen values.
h) If ? is an eigen value of A, show that ?
?1
is an eigen value of A
?1
.
i) Check whether the vectors u
12
3
5
? ?
? ?
?
? ?
? ? ?
? ?
and v
2
3
3
? ?
? ?
? ?
? ?
? ?
? ?
are orthogonal or not?
j) The characteristic roots of
8 6 2
6 4
2 4 3
A k
? ? ?
? ?
? ? ?
? ?
? ?
?
? ?
are 0, 3, 15. Find the value of k.

SECTION-B
2. a) Determine if the following system is consistent :
y ? 4z = 8
2x ? 3y + 2z = 1
4x ? 8y + 12z = 1
b) Let u =
1 2 4
4 , 3 and 1
2 7
v w
h
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
, For what value(s) of h is w in the plane spanned
by u and v ?
3. a) Given
1 2 4
0 1 5
2 4 3
A
? ?
? ?
?
? ?
? ?
? ? ?
? ?
and
2
2
9
b
? ? ?
? ?
?
? ?
? ?
? ?
, write the augmented matrix for the linear
system that corresponds to the matrix equation Ax = b. Then solve the system and
write the solution as a vector.
b) Find the inverse of the matrix
0 1 2
1 0 3
4 3 8
? ?
? ?
? ?
? ?
?
? ?
using row transformations.
4. Let T : R
3
? R
3
be a linear transformation defined by
0 x
T y x y
z x y z
? ? ? ?
? ? ? ?
? ?
? ? ? ?
? ? ? ?
? ?
? ? ? ?
. Find the
matrix representation of T w.r.t. the ordered basis B
1
= {(1, 0, 0), (0, 1, 0), (0, 0, 1)} and
B
2
= {(1, 0, 1), (1, 1, 0), (0, 1, 1)}.
3 | M-77256 (S1)-2591

5. a) Let v
1
= (1, ?1, 0), v
2
= (0, 1, ?1) and v
3
= (0, 0, 1) be elements of R
3
. Show that the
set of vectors {v
1
, v
2
, v
3
} is linearly independent.
b) Prove that
2
2
2
1
1 0
1
w w
w w
w w
? , where w is a cube root of unity.

SECTION-C
6. a) Let
1 2
2 4
3 , 5 ,
5 8
u u
? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
and
8
2
9
v
? ?
? ?
?
? ?
? ? ?
? ?
. Determine whether v is in the subspace of R
3

generated by u
1
and u
2
.
b) Solve the following system of linear equations by Cramer?s rule :
x + y + z = 6, x ? y + 2z = 5, 3x + y + z = 8
7. Determine the eigen values and corresponding eigen vectors of the matrix
6 2 2
2 3 1
2 1 3
? ? ?
? ?
? ?
? ?
? ?
?
? ?
.
8. Diagonalize the matrix
1 6 1
1 2 0
0 0 3
? ?
? ?
? ?
? ?
? ?
.
9. Find an orthogonal basis or the coloumn space of the matrix
3 5 1
1 1 1
1 5 3
3 7 8
? ? ?
? ?
? ?
? ? ?
? ?
?
? ?





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