Intermediate Part FIRST Series Test#2CHAPTER NO.

3
MULTIPLE CHOICE QUESTIONS
i. If the matrix has only one row but more than one column, the matrix is
(a) Column (b) Real (c) Row (d) None of these

ii. The square matrix A is diagonal matrix if

(a) a ij=0 ∀ i ≠ j (b) a ij=0 ∀ . i≠ j
(c) a ij=0 ∀ i ≠ j and at least one a ij ≠ 0 ∀ . i= j (d) All a ij=0 ∀ i= j

iii. If the order of A is m × p and order if B is p ×n then order of AB is.

(a) m ×n (b) p ×m (c) n × m (d) n × p

iv. If AB=BA=I then both A and B are

(a) Additive inverse of each other (b) Multiplicative inverse of each other
(c) Equal matrices (d) Singular matrices

v. A square matrix A=[aij ]3 ×3 is called upper triangular if

(a) a ij=0 ,i ≠ j (b) a ij ≠ 0 ,i< j (c) a ij ≠ 0 ,i> j (d) a ij=0 ,i= j

vi. A square matrix A=[aij ]3 ×3 is called skew symmetric if

(a) A=A (b) A=−A (c) A=IA (d) None of these

vii. A equation of the form ax +by =c is called

(a) Linear equation (b) Homogeneous linear equation
(c) Non-homogeneous linear equation (d) None of these

viii. If the system has no solution, then the system is

(a) Consistent (b) Inconsistent (c) Unique (d) None of these

ix. If the co-efficient matrix of the homogenous system is singular then the solution is
(a) Non-trivial (b)Trivial (c) Does not exist (d) None of these

| |
rcos ∅ 1 −sin ∅
x. 0 1 0 =¿
rsin ∅ 0 cos ∅
(a) 1 (b) r (c) r 2 (d) r 3

xi. If ( A)t =A then A is_________ matrix:

(a) Symmetric (b) Skew-symmetric (c) Hermitian (d) Skew-hermitian

xii. If [ 1x 10] and A2=I , then x is equal to:

(a) 0 (b) 1 (c) 3 (d) 2

[ ]
1 −2 3
xiii. If A= −2 3 1 then A33 equals.
4 −3 2
(a) -1 (b) 1 (c) 3 (d) 2

xiv. If ¿ A∨¿ is the determinant of a square matrix A then ¿ A∨¿is:

(a) Always positive (b) Modulus of A (c) Always negative (d) May be + ve or — ve

xv. If 'A' is a square matrix of order 3 x 3 then |KA| equals:

(a) k ∨ A∨¿ (b)| A| (c) k3 IA| (d) k4|A|
 xvi. For matrix equation [−33 3 y−4
1
]=[−33 12] the value of y=¿

(a) 4 (b) 1 (c) 3 (d) 2

xvii. (AB)t =
(a) At (b) Bt (c) Bt At (d) At Bt

xviii. If ‘A’ is a square matrix of order 3 ×3 then ¿ A∨¿

(a) a11A11 + a12A12 + a13A13 (b) a11A11−¿ a12 A12−¿ a13 A13
(c) A11 A12 A13 (d) a11 + a12 + a13

xix. If ‘A’ is multiplicative inverse of 'B' then:

(a) AB = I (b) BA = I (c) Both (a) and (b) (d) None of these

| |
1 23 x
xx. 2 3 6 x =?
3 49 x
(a) 1 (b) (3x + 1)(x - 1) (c) 0 (d) x3
Q No. 2. Write at least Eight (8) short answers of the following questions.
i. Define multiplication of matrices.

ii. Define transpose of a matrix.

iii. Find x and y if [ x+3 =

[
−3 3 y−4 −3 2 x ]
1
] y 1
.

iv. If A=[aij ]3 ×3 show that|KA|=K 3| A|

v. If A = [ 1a 2b]∧ A =[ 00 00]
2
find the values of a , b .

[ ][ ]
r cos ∅ 0 −sin ∅ r cos ∅ 0 −sin ∅
vi. Show 0 r 0 0 1 0 =r I 3
r sin ∅ 0 cos ∅ r sin ∅ 0 cos ∅

vii. Find matrix x if X [−25 21]=[−112 53]

viii. If A=[−4 5]
−2 3
, find inverse.

ix. Solve the following system of linear equations 2 x1 −3 x 2=5 ; 5 x 1−x 2=4

x. If A and B are square matrices of same order, then explain why in general. (A+B)2 ≠ A2 + 2AB + B2

xi. Solve the following matrix equations for X: 3 X −2 A=B if A= [−12 31−25] , B=[ 25−34−11 ]
xii. Solve the following matrix equation for A. [ 42 32] A−[−12 −23 ]=[−13−46 ]
Q No. 2. Write at least Eight (8) short answers of the following questions.

| |
2a a a
i. Evaluate b 2 b b
c c 2c

| |
2 3 −1
ii. Without expansion show that 1 1 0 =0
2 −3 5

| || |
230 210
iii. Show that 396 =9 112
2 15 1 251

| |
b +c a a
iv. Show that b c+ a b =4 abc
c c a+b

| |
a+ λ b c 2
v. Show that a b+ λ c =λ ( a+b+ c+ λ)
a b c+ λ
 ||
a
2 bc
1a
b =0
vi. Without expansion verify that 1 b2
ac
1 c2
c
ab

| || |
2 2 3
mn I l 1l l
vii. Without expansion verify that nl m m 2 = 1 m2 m 3
lm n n2 1 n 2 n3

| |
2 a 2b 2 c
viii. Without expansion verify that a+b 2b b+ c =0
a+c b+ c 2 c

| |
12 1
ix. Find value of x if 2 x 2 =0
36 x

| |
4 λ3
x. ’
Find value of ‘λ if A is singular A= 7 36
2 31

xi. Verify that ( A−1)t =( A t )−1 if A=¿ [ 23−11]

xii. If A and B are non- singular matrices, then show that ( A−1)−1= A
Q.4 Write at least Nine (9) short answers of the following questions.
i. Write any four properties of determinant.

ii. Define skew – symmetric matrix.

iii. Define hermitian matrix.

iv. Define Rank of matrix.

[ ]
1 2 0
v. If A = 3 2 −1 show that A−A t is skew symmetric
−1 3 2

vi. If A is any square matrix of order 3, show that A+ A t is symmetric.

vii. If A is symmetric or skew symmetric show that A2 is symmetric.

{
x +2 y – 2 z =0
viii. Solve the following system of homogeneous linear equations. 2 x+ y+ 5 z =0
5 x + 4 y +8 z=0

[ ] [ ]
1 −2 5 −3 1 −2
ix. If A= −2 3 −1 , B= 1 0 −1 then show that A+ B is symmetric.
−5 −1 0 −2 −1 2

x. Show that A At is symmetric for any matrix of order 2 × 3.

| |
1 2 −3
xi. Evaluate −1 3 4
−2 5 6

xii. If [ 1i −i0 ] , Show that A4 = I2

xiii. Define Echelon form of a matrix.

SECTION-II
Note: Attempt any three (3) questions. (10 x 3 = 30)

[ ]
1 1 3
−1
Q No 5 (a) Find A if A = 5 2 6
−2−1 −3

[ ]
2 −1 3 0
(b) If A= 1 0 4 −2 then find A At and At A
−3 5 2 −1

| |
−a 0 c
Q No 6 (a) without expansion verify that 0 a −b =0
b −c 0

| |
34 2 7
25 0 3
(b) Evaluate =0
1 2 −3 5
4 1 −2 6

{
2 x +2 y + z=13
Q No 7 (a) Solve the following systems of linear equations by Cramer rule. 3 x – 2 y – 2 z=1
5 x + y−3 z=2
 [ ]
1 2 −3
(b) Find the inverse of A= 0 −2 0 by using row operation
−2 −2 2

[ ]
1 −1 2 1
Q No 8 (a) Find the rank of A= 2 −6 5 1
3 5 4 −3
(b) Solve the systems by reducing their augmented matrices into echelon form.

{
x1 −2 x 2−2 x 3=−1
2 x 1+3 x 2 + x 3=1
5 x 1−4 x 2−3 x 3 =1

Q No 9 (a) Verify that ( AB)−1=B−1 A−1 if A= [−11 20] B=[−34−11 ]

[ ][ ]
r cos ∅ 0 −sin ∅ r cos ∅ 0 −sin ∅
(b) Show that 0 r 0 0 1 0 =¿ r I3
r sin ∅ 0 cos ∅ r sin ∅ 0 cos ∅