Linear-Algebra

Question 1

Let A and B be real symmetric matrices of size n × n. Then which one of the following is true?

A
AA′ = 1
B
A = A-1
C
AB = BA
D
(AB)' = BA
Question 1 Explanation: 
Question 2

The tank of matrix is:

A
0
B
1
C
2
D
3
Question 2 Explanation: 
Question 3

In a compact single dimensional array representation for lower triangular matrices (i.e all the elements above the diagonal are zero) of size n × n, non-zero elements (i.e elements of the lower triangle) of each row are stored one after another, starting from the first row, the index of the (i, j)th element of the lower triangular matrix in this new representation is:

A
i + j
B
i + j - 1
C
j + i(i-1)/2
D
i + j(j-1)/2
Question 3 Explanation: 
Though not mentioned in question, from options it is clear that array index starts from 1 and not 0.
If we assume array index starting from 1 then, ith row contains i number of non-zero elements. Before ith row there are (i-1) rows, (1 to i-1) and in total these rows has 1+2+3......+(i-1) = i(i-1)/2 elements.
Now at ith row, the jth element will be at j position.
So the index of (i, j)th element of lower triangular matrix in this new representation is
j = i(i-1)/2
Question 4

Find the inverse of the matrix

A
B
C
D
Question 4 Explanation: 
Using eigen values, the characteristic equation we get is,
3 + 2λ2 - 2 = 0
Using Cayley-Hamiltonian theorem
-A3 + 2A2 - 2I = 0
So, A-1 = 1/2 (2A - A2)
Solving we get,
Question 5

In the interval [0, π] the equation x = cos x has

A
No solution
B
Exactly one solution
C
Exactly two solutions
D
An infinite number of solutions
Question 5 Explanation: 

x & cos(x) are intersecting at only one point.
Question 6

The rank of the following (n + 1)×(n+1) matrix, where a is a real number is

A
1
B
2
C
n
D
Depends on the value of a
Question 6 Explanation: 
Question 7

Let A be the set of all non-singular matrices over real number and let* be the matrix multiplication operation. Then

A
A is closed under* but < A, *> is not a semigroup
B
is a semigroup but not a monoid
C
is a monoid but not a group
D
is a group but not an abelian group
Question 7 Explanation: 
As the matrices are non-singular so their determinant ±0. Hence, the inverse matrix always exist. But for a group to be Abelian it should follow commutative property. As matrix multiplication is not commutative, is a group but not an abelian group.
Question 8

Let A and B be two n×n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements,

    I. rank(AB) = rank(A) rank(B)
    II. det(AB) = det(A) det(B)
    III. rank(A + B) ≤ rank(A) + rank(B)
    IV. det(A + B) ≤ det(A) + det(B)

Which of the above statements are TRUE?

A
I and II only
B
I and IV only
C
III and IV only
D
II and III only
Question 8 Explanation: 
Rank(AB) ≥ Rank(A) + Rank(B) − n. So option I is wrong.
Rank is the number of independent rows(vectors) of a matrix. On product of two matrices, the combined rank is more than the sum of individual matrices (subtracted with the order n)
det(AB) = det(A)∙det(B) as the magnitude remains same for the matrices after multiplication.
Note: We can just take a 2x2 matrix and check the options.
Question 9

Let Ax = b be a system of linear equations where A is an m × n matrix and b is a m × 1 column vector and X is a n × 1 column vector of unknowns. Which of the following is false?

A
The system has a solution if and only if, both A and the augmented matrix [A b] have the same rank.
B
If m < n and b is the zero vector, then the system has infinitely many solutions.
C
If m = n and b is non-zero vector, then the system has a unique solution.
D
The system will have only a trivial solution when m = n, b is the zero vector and rank (A) = n.
Question 9 Explanation: 
→ It belongs to linear non-homogeneous equations. So by having m=n, we can't say that it will have unique solution.
→ Solution can be depends on rank of matrix A and matrix [A B].
→ If rank[A] = rank[A B] then it can have solution otherwise no solution.
Question 10

The matrices and commute under multiplication

A
if a = b or θ = nπ, is an integer
B
always
C
never
D
if a cos θ ≠ b sin θ
Question 10 Explanation: 
Question 11

Let and be two matrices such that AB = I. Let and CD = 1. Express the elements of D in terms of the elements of B.

A
Theory Explanation.
Question 12

The determinant of the matrix is is:

A
11
B
-48
C
0
D
-24
Question 12 Explanation: 
Determinant of given matrix = 6 × 2 × 4 × (-1) = -48
Question 13

Let A = (aij) be an n-rowed square matrix and I12 be the matrix obtained by interchanging the first and second rows of the n-rowed Identify matrix. Then AI12 is such that its first

A
row is the same as its second row
B
row is the same as the second row of A
C
column is the same as the second column of A
D
row is all zero
Question 13 Explanation: 
Let A be 3×3 matrix and I12 be matrix obtained by interchanging the first and second rows of the 3-rowed Identity matrix.

So, we can see that column 1 and 2 got interchanged.
Question 14

Consider the following set of equations

                x + 2y = 5
               4x + 8y = 12
          3x + 6y + 3z = 15 

This set

A
has unique solution
B
has no solutions
C
has finite number of solutions
D
has infinite number of solutions
Question 14 Explanation: 
Question 15

The rank of the matrix given below is:

        1   4   8   7
        0   0   3   0
        4   2   3   1
        3   12  24  2  
A
3
B
1
C
2
D
4
Question 15 Explanation: 
Question 16

Consider the following determinant 

Which of the following is a factor of Δ?

A
a+b
B
a-b
C
a+b+c
D
abc
Question 16 Explanation: 
Question 17

Derive the expression for the number of expressions required to solve a system of linear equations in n unknowns using the Gaussian Elimination Method. Assume that one operation refers to a multiplication followed by an addition.

A
Theory Explanation.
Question 18

An n x n array v is defined as follows:

v[i,j] = i-j for all i,j, 1 ≤ i ≤ n, 1 ≤ j ≤ n 

The sum of the elements of the array v is

A
0
B
n -1
C
n2 - 3n + 2
D
n2 (n+1)/2
Question 18 Explanation: 
Let us consider n=5 then we get

Add ith row and jth column if we zero, apply to all row and their corresponding column the total becomes zero.
Question 19

The determinant of the matrix is

is:

A
4
B
0
C
15
D
20
Question 19 Explanation: 
The value of the determinant is 2 * 1 * 2 * 1 = 4
Question 20

Consider the following statements:

    S1: The sum of two singular n × n matrices may be non-singular
    S2: The sum of two n × n non-singular matrices may be singular.

Which of the following statements is correct?

A
S1 and S2 are both true
B
S1 is true, S2 is false
C
S1 is false, S2 is true
D
S1 and S2 are both false
Question 20 Explanation: 
Question 21

The rank of the matrix is

A
4
B
2
C
1
D
0
Question 21 Explanation: 
Number of non-zero rows is the rank of the matrix.
Question 22

Consider the following system of linear equation

Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of α, does this system of equations have infinitely many solutions?

A
0
B
1
C
2
D
infinitely many
Question 22 Explanation: 

This is in the form AX = B

⇒ R(AB) < n [If we want infinitely many solution]
then -1+5α = 0
5α = 1
α = 1/5 There is only one value of α. System can have infinitely many solutions.
Question 23

The number of different n × n symmetric matrices with each element being either 0 or 1 is: (Note: power(2,x) is same as 2x)

A
power (2,n)
B
power (2,n2)
C
power (2, (n2 + n)/2)
D
power (2, (n2 - n)/2)
Question 23 Explanation: 
If a matrix is symmetric then
A[i][j] = A[j][i]
So, we have only two choices, they are either upper triangular elements (or) lower triangular elements.
No. of such elements are
n + (n-1) + (n-2) + ... + 1
n(n+1)/2
We have two choices, thus we have
2(n(n+1)/2) = 2((n2+n)/2) choices
i.e., Power (2, (n2+n)/2).
Question 24

Let A, B, C, D be n × n matrices, each with non-­zero determinant. If ABCD = 1, then B-1 is

A
D-1C-1A-1
B
CDA
C
ADC
D
Does not necessarily exist
Question 24 Explanation: 
ABCD are n × n matrices with non-zero determinant.
ABCD = I
Pre multiply A-1 on both sides
A-1ABCD = A-1⋅I
BCD = A-1
Pre multiply B-1 on both sides
B-1BCD = B-1A-1
CD = B-1A-1
Post multiply A on both sides
CDA = B-1A-1⋅A
∴ CDA = B-1(I)
∴ CDA = B-1
Question 25

How many solutions does the following system of linear equations have?

  -x + 5y = -1
   x - y = 2
   x + 3y = 3 
A
infinitely many
B
two distinct solutions
C
unique
D
none
Question 25 Explanation: 

rank = r(A) = r(A|B) = 2
rank = total no. of variables
Hence, unique solution.
Question 26

In an M×N matrix such that all non-zero entries are covered in a rows and b columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is

A
≤ a+b
B
≤ max(a, b)
C
≤ min(M-a, N-b)
D
≤ min(a, b)
Question 26 Explanation: 
Entry will be a member of same row and same column.
→ Such that a row can have maximum of a elements and no row has separate element and for b also same.
→ By combining the both, it should be ≤ (a,b).
Question 27

The eigen vector(s) of the matrix

is (are)

A
(0,0,α )
B
(α,0,0)
C
(0,0,1)
D
(0,α,0)
E
Both B and D
Question 27 Explanation: 
Since, the given matrix is an upper triangular one, all eigen values are A. And hence A - λI = A.
So the question as has
(A - λI)X = 0
AX = 0

What x1, x2, x3 are suitable?
Which means:
x1 times column 1 + x2 times column 2 + x3 times column 3 = zero vector
Since α is not equal to zero, so x3 must be necessarily zero to get zero vector.
Hence, only (B) and (D) satisfies.
Question 28

If the matrix A4, calculated by the use of Cayley-Hamilton theorem or otherwise, is _________

A
A4 = I
Question 28 Explanation: 
Let λ be eigen value, then characteristic equation will be
(1-λ) (-1-λ) (i-λ) (-i-λ)
= (λ2-1) (λ2+1)
= λ4-1
Characteristic equation is λ4-1 = 0.
According to Cayley-Hamilton theorem, every matrix satisfies its own characteristic equation, so
A4 = I
Question 29

The number of integer-triples (i.j.k) with 1 ≤ i.j.k ≤ 300 such that i + j + k is divisible by 3 is ________

A
Fill in the blanks.
Question 30
A
3
Question 30 Explanation: 
Question 31
. Suppose that P is a 45matrix such that every solution of the equation Px=0 is a scalar multiple of [2  5  4  3  1]T. The rank of P is _________.
A
4
Question 31 Explanation: 
If the rank of a homogeneous system is less than the number of variables in the system, then the system has infinitely many solutions. r
Question 32
A
There exists a bijection from S1to S2.
B
There does not exist a bijection from S1to S2.
C
There exists a surjection from S1to S2.
D
There does not exist an injunction from S1to S2.
Question 32 Explanation: 

The number of functions from a set A to set B is |B|^|A|.

S2: |B|= 3, |A|= n^2-1 +1 = n^2.
we have number of functions 3^(n^2).

S1: there are n*n positions in a matrix of size nxn. Each can be filled with either 0 or 1 or 2 i,e, in 3^(n^2)

 

As there are equal number of elements on both sides, S1->S2 can be one one , onto as well bijection



Question 33

The determinant of the matrix given below is

A
-1
B
0
C
1
D
2
Question 33 Explanation: 

determinant = product of diagonal element [upper triangular matrix]
= -1 * 1 * 1 * 1
= -1
Question 34

What values of x, y and z satisfy the following system of linear equations?

A
x = 6, y = 3, z = 2
B
x = 12, y = 3, z = -4
C
x = 6, y = 6, z = -4
D
x = 12, y = -3, z = 0
Question 34 Explanation: 
Question 35

Let A be an Let A be an n × n matrix of the following form.

What is the value of the determinant of A?

A
B
C
D
Question 35 Explanation: 
Put n=1, you will get a matrix like [3].
Find its determinant, Determinant = 3.
Now check options, by putting n=1, I am getting following results,
A) 5
B) 7
C) 3
D) 3
(A), (B) can't be the answer.
Now, check for n=2, Determinant = 9-1 = 8.
Put n=2 in (C), (D)
C) 7
D) 8
So, (D) is the answer.
Question 36

Let H1, H2, H3, ... be harmonic numbers. Then, for n ∈ Z+, can be expressed as

A
nHn+1 – (n + 1)
B
(n + 1)Hn – n
C
(n + 1)Hn – n
D
(n+1)Hn+1 – (n+1)
Question 37

If matrix and X2 - X + I = 0 (I is the Identity matrix and 0 is the zero matrix), then the inverse of X is:

A
B
C
D
Question 37 Explanation: 
Question 38

What are the eigenvalues of the matrix P given below

A
B
a, a, a
C
0, a, 2a
D
-a, 2a, 2a
Question 38 Explanation: 
There are 38 questions to complete.

Access quiz wise question and answers by becoming as a solutions adda PRO SUBSCRIBER with Ad-Free content

Register Now

If you have registered and made your payment please contact solutionsadda.in@gmail.com to get access