## Linear-Algebra

 Question 1
 A 3
Question 1 Explanation:
 Question 2

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 2 Explanation:
 Question 3

The tank of matrix is:

 A 0 B 1 C 2 D 3
Question 3 Explanation:
 Question 4

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 4 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 5

Find the inverse of the matrix

 A B C D
Question 5 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 6

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 6 Explanation:

x & cos(x) are intersecting at only one point.
 Question 7

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 7 Explanation:
 Question 8

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 8 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 9

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 9 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 10

Let A be the 2×2 matrix with elements a11 = a12 = a21 = +1 and a22 = -1. Then the eigenvalues of the matrix A19 are

 A 1024 and -1024 B 1024√2 and -1024√2 C 4√2 and -4√2 D 512√2 and -512√2
Question 10 Explanation:
a11 = a12 = a21 = 1, a22 = -1
The 2×2 matrix =
Cayley Hamilton theorem:
If matrix A has ‘λ’ as eigen value, An has eigen value as λn.
Eigen value of
|A-λI| = 0

-(1-λ)(1+λ)-1 = 0
-(1-λ2 )-1 = 0
-1 = 1-λ2
λ2 = 2
λ = ±√2
A19 has (√2)19 = 29×√2 (or) (-√2)19 = -512√2
= 512√2
 Question 11

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 11 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 12

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 12 Explanation:
 Question 13

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 14

The determinant of the matrix is is:

 A 11 B -48 C 0 D -24
Question 14 Explanation:
Determinant of given matrix = 6 × 2 × 4 × (-1) = -48
 Question 15

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 15 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 16

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 16 Explanation:
 Question 17

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 17 Explanation:
 Question 18

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 18 Explanation:
 Question 19

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 20

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 20 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 21

The determinant of the matrix is

is:

 A 4 B 0 C 15 D 20
Question 21 Explanation:
The value of the determinant is 2 * 1 * 2 * 1 = 4
 Question 22

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 22 Explanation:
 Question 23

The rank of the matrix is

 A 4 B 2 C 1 D 0
Question 23 Explanation:
Number of non-zero rows is the rank of the matrix.
 Question 24

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 24 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 25

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 25 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 26

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

 A A4 = I
Question 26 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 27

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 28
. 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 28 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 29
 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 29 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 30

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 30 Explanation:
 Question 31

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 31 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
 Question 32

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 33

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 33 Explanation:
 Question 34

The determinant of the matrix given below is

 A -1 B 0 C 1 D 2
Question 34 Explanation:

determinant = product of diagonal element [upper triangular matrix]
= -1 * 1 * 1 * 1
= -1
 Question 35

What are the eigenvalues of the matrix P given below

 A B a, a, a C 0, a, 2a D -a, 2a, 2a
Question 35 Explanation:
 Question 36

Consider the following set of linear operations:
x + y/2 = 9
3x + y = 10
The value of the Frobenius norm for the above system of equations is:

 A 0.5 B 0.75 C 1.5 D 2
Question 36 Explanation:
 Question 37

Consider the following set of linear operations:
x + y/2 = 9
3x + y = 10
What can be said about the Gauss-Siedel iterative method for solving the above set of linear equations?

 A it will converge B it will diverse C it will neither converge nor diverse D It is not applicable
Question 37 Explanation:
As,
|1| + |1/2| <= |9|
and |3| + |1| <= |10|
 Question 38

Let A be the What is the maximum value of xTAx where the maximum is taken over all x that are the unit eigenvectors of A?

 A 5 B 5+√5/2 C 3 D 5-√5/2
Question 38 Explanation:
There are 38 questions to complete.

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