LinearAlgebra
Question 1 
3 
Question 2 
Let A and B be real symmetric matrices of size n × n. Then which one of the following is true?
AA′ = 1  
A = A^{1}  
AB = BA  
(AB)' = BA 
Question 3 
The tank of matrix is:
0  
1  
2  
3 
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, nonzero 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:
i + j  
i + j  1  
j + i(i1)/2  
i + j(j1)/2 
If we assume array index starting from 1 then, i^{th} row contains i number of nonzero elements. Before i^{th} row there are (i1) rows, (1 to i1) and in total these rows has 1+2+3......+(i1) = i(i1)/2 elements.
Now at i^{th} row, the j^{th} 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(i1)/2
Question 5 
Find the inverse of the matrix
λ^{3} + 2λ^{2}  2 = 0
Using CayleyHamiltonian theorem
A^{3} + 2A^{2}  2I = 0
So, A^{1} = 1/2 (2A  A^{2})
Solving we get,
Question 6 
In the interval [0, π] the equation x = cos x has
No solution  
Exactly one solution  
Exactly two solutions  
An infinite number of solutions 
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
1  
2  
n  
Depends on the value of a 
Question 8 
Let A be the set of all nonsingular matrices over real number and let* be the matrix multiplication operation. Then
A is closed under* but < A, *> is not a semigroup  
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?
I and II only
 
I and IV only  
III and IV only  
II and III only

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 a_{11} = a_{12} = a_{21} = +1 and a_{22} = 1. Then the eigenvalues of the matrix A^{19} are
1024 and 1024  
1024√2 and 1024√2  
4√2 and 4√2  
512√2 and 512√2 
The 2×2 matrix =
Cayley Hamilton theorem:
If matrix A has ‘λ’ as eigen value, A^{n} has eigen value as λ^{n}.
Eigen value of
AλI = 0
(1λ)(1+λ)1 = 0
(1λ^{2} )1 = 0
1 = 1λ^{2}
λ^{2} = 2
λ = ±√2
A^{19} has (√2)^{19} = 2^{9}×√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?
The system has a solution if and only if, both A and the augmented matrix [A b] have the same rank.
 
If m < n and b is the zero vector, then the system has infinitely many solutions.  
If m = n and b is nonzero vector, then the system has a unique solution.  
The system will have only a trivial solution when m = n, b is the zero vector and rank (A) = n. 
→ 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
if a = b or θ = nπ, is an integer  
always  
never  
if a cos θ ≠ b sin θ 
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.
Theory Explanation. 
Question 14 
The determinant of the matrix is is:
11  
48  
0  
24 
Question 15 
Let A = (a_{ij}) be an nrowed square matrix and I_{12} be the matrix obtained by interchanging the first and second rows of the nrowed Identify matrix. Then AI_{12} is such that its first
row is the same as its second row  
row is the same as the second row of A  
column is the same as the second column of A  
row is all zero 
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
has unique solution  
has no solutions  
has finite number of solutions  
has infinite number of solutions 
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
3  
1  
2  
4 
Question 18 
Consider the following determinant
Which of the following is a factor of Δ?
a+b  
ab  
a+b+c  
abc 
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.
Theory Explanation. 
Question 20 
An n x n array v is defined as follows:
v[i,j] = ij for all i,j, 1 ≤ i ≤ n, 1 ≤ j ≤ n
The sum of the elements of the array v is
0  
n 1  
n^{2}  3n + 2  
n^{2} (n+1)/2 
Add i^{th} row and j^{th} 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:
4  
0  
15  
20 
Question 22 
Consider the following statements:
 S1: The sum of two singular n × n matrices may be nonsingular
S2: The sum of two n × n nonsingular matrices may be singular.
Which of the following statements is correct?
S1 and S2 are both true  
S1 is true, S2 is false  
S1 is false, S2 is true  
S1 and S2 are both false 
Question 23 
The rank of the matrix is
4  
2  
1  
0 
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?
0  
1  
2  
infinitely many 
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)
(0,0,α )  
(α,0,0)  
(0,0,1)  
(0,α,0)  
Both B and D 
So the question as has
(A  λI)X = 0
AX = 0
What x_{1}, x_{2}, x_{3} are suitable?
Which means:
x_{1} times column 1 + x_{2} times column 2 + x_{3} times column 3 = zero vector
Since α is not equal to zero, so x_{3} must be necessarily zero to get zero vector.
Hence, only (B) and (D) satisfies.
Question 26 
If the matrix A^{4}, calculated by the use of CayleyHamilton theorem or otherwise, is _________
A^{4} = I 
(1λ) (1λ) (iλ) (iλ)
= (λ^{2}1) (λ^{2}+1)
= λ^{4}1
Characteristic equation is λ^{4}1 = 0.
According to CayleyHamilton theorem, every matrix satisfies its own characteristic equation, so
A^{4} = I
Question 27 
The number of integertriples (i.j.k) with 1 ≤ i.j.k ≤ 300 such that i + j + k is divisible by 3 is ________
Fill in the blanks. 
Question 28 
4 
Question 29 
There exists a bijection from S_{1}to S_{2}.  
There does not exist a bijection from S_{1}to S_{2}.  
There exists a surjection from S_{1}to S_{2}.  
There does not exist an injunction from S_{1}to S_{2}. 
The number of functions from a set A to set B is B^A.
S2: B= 3, A= n^21 +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?
x = 6, y = 3, z = 2  
x = 12, y = 3, z = 4  
x = 6, y = 6, z = 4  
x = 12, y = 3, z = 0 
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?
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 = 91 = 8.
Put n=2 in (C), (D)
C) 7
D) 8
So, (D) is the answer.
Question 32 
Let H_{1}, H_{2}, H_{3}, ... be harmonic numbers. Then, for n ∈ Z^{+}, can be expressed as
nH_{n+1} – (n + 1)  
(n + 1)H_{n} – n  
(n + 1)H_{n} – n  
(n+1)H_{n+1} – (n+1) 
Question 33 
If matrix and X_{2}  X + I = 0 (I is the Identity matrix and 0 is the zero matrix), then the inverse of X is:
Question 34 
The determinant of the matrix given below is
1  
0  
1  
2 
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, a, a  
0, a, 2a  
a, 2a, 2a 
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:
0.5  
0.75  
1.5  
2.0 
Question 37 
Consider the following set of linear operations:
x + y/2 = 9
3x + y = 10
What can be said about the GaussSiedel iterative method for solving the above set of linear equations?
it will converge  
it will diverse  
it will neither converge nor diverse  
It is not applicable 
1 + 1/2 <= 9
and 3 + 1 <= 10
Question 38 
Let A be the What is the maximum value of x^{T}Ax where the maximum is taken over all x that are the unit eigenvectors of A?
5  
5+√5/2  
3  
5√5/2 