LinearAlgebra
Question 1 
I. X is invertible.
II. Determinant of X is nonzero.
Which one of the following is TRUE?
I implies II; II does not imply I.  
II implies I; I does not imply II.  
I and II are equivalent statements.  
I does not imply II; II does not imply I. 
That means we can also say that determinant of X is nonzero.
Question 2 
Consider the following matrix:
The absolute value of the product of Eigen values of R is ______.
12  
17  
10  
8 
Question 3 
The largest eigenvalue of A is ________
3  
4  
5  
6 
→ Correction in Explanation:
⇒ (1  λ)(2  λ)  2 = 0
⇒ λ^{2}  3λ=0
λ = 0, 3
So maximum is 3.
Question 4 
Assume that multiplying a matrix G_{1} of dimension p×q with another matrix G_{2} of dimension q×r requires pqr scalar multiplications. Computing the product of n matrices G_{1}G_{2}G_{3}…G_{n} can be done by parenthesizing in different ways. Define G_{i}G_{i+1} as an explicitly computed pair for a given parenthesization if they are directly multiplied. For example, in the matrix multiplication chain G_{1}G_{2}G_{3}G_{4}G_{5}G_{6} using parenthesization(G_{1}(G_{2}G_{3}))(G_{4}(G_{5}G_{6})), G_{2}G_{3} and G_{5}G_{6} are the only explicitly computed pairs.
Consider a matrix multiplication chain F_{1}F_{2}F_{3}F_{4}F_{5}, where matrices F_{1}, F_{2}, F_{3}, F_{4} and F_{5} are of dimensions 2×25, 25×3, 3×16, 16×1 and 1×1000, respectively. In the parenthesization of F_{1}F_{2}F_{3}F_{4}F_{5} that minimizes the total number of scalar multiplications, the explicitly computed pairs is/ are
F_{1}F_{2} and F_{3}F_{4} only
 
F_{2}F_{3} only  
F_{3}F_{4} only  
F_{1}F_{2} and F_{4}F_{5} only

→ Optimal Parenthesization is:
((F_{1}(F_{2}(F_{3} F_{4})))F_{5})
→ But according to the problem statement we are only considering F_{3}, F_{4} explicitly computed pairs.
Question 5 
Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (H), medium (M) and low (L). Let P(H_{G}) denote the probability that Guwahati has high temperature. Similarly, P(M_{G}) and P(L_{G}) denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use P(H_{D}), P(M_{D}) and P(L_{D}) for Delhi.
The following table gives the conditional probabilities for Delhi’s temperature given Guwahati’s temperature.
Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature (H_{G}) then the probability of Delhi also having a high temperature (H_{D}) is 0.40; i.e., P(H_{D} ∣ H_{G}) = 0.40. Similarly, the next two entries are P(M_{D} ∣ H_{G}) = 0.48 and P(L_{D} ∣ H_{G}) = 0.12. Similarly for the other rows.
If it is known that P(H_{G}) = 0.2, P(M_{G}) = 0.5, and P(L_{G}) = 0.3, then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is _______ .
0.60  
0.61  
0.62  
0.63 
The first entry denotes that if Guwahati has high temperature (H_{G} ) then the probability that Delhi also having a high temperature (H_{D} ) is 0.40.
P (H_{D} / H_{G} ) = 0.40
We need to find out the probability that Guwahati has high temperature.
Given that Delhi has high temperature (P(H_{G} / H_{D} )).
P (H_{D} / H_{G} ) = P(H_{G} ∩ H_{D} ) / P(H_{D} )
= 0.2×0.4 / 0.2×0.4+0.5×0.1+0.3×0.01
= 0.60
Question 6 
P: Set of Rational numbers (positive and negative)
Q: Set of functions from {0, 1} to N
R: Set of functions from N to {0, 1}
S: Set of finite subsets of N
Which of the above sets are countable?
Q and S only  
P and S only  
P and R only  
P, Q and S only 
Set of functions from {0,1} to N is countable as it has one to one correspondence to N.
Set of functions from N to {0,1} is uncountable, as it has one to one correspondence to set of real numbers between (0 and 1).
Set of finite subsets of N is countable.
Question 7 
Consider the following statements.
(I)P does not have an inverse
(II)P has a repeated eigenvalue
(III)P cannot be diagonalized
Which one of the following options is correct?
Only I and III are necessarily true  
Only II is necessarily true  
Only I and II are necessarily true  
Only II and III are necessarily true 
Though the multiple of a vector represents same vector, and each eigen vector has distinct eigen value, we can conclude that ‘p’ has repeated eigen value.
If the unique eigen value corresponds to an eigen vector e, but the repeated eigen value corresponds to an entire plane, then the matrix can be diagonalized, using ‘e’ together with any two vectors that lie in plane.
But, if all eigen values are repeated, then the matrix cannot be diagonalized unless it is already diagonal.
So (III) holds correct.
A diagonal matrix can have inverse, So (I) is false.
Then (II) and (III) are necessarily True.
Question 8 
Let c^{1}, c^{n} be scalars not all zero. Such that the following expression holds:
where a_{i} is column vectors in R^{n}. Consider the set of linear equations.
Ax = B.where A = [a_{1}.......a_{n}] and
Then, Set of equations has
a unique solution at x = J_{n} where J_{n} denotes a ndimensional vector of all 1  
no solution  
infinitely many solutions  
finitely many solutions 
AX = B
As given that
and c_{1}&c_{n} ≠ 0
means c_{0}a_{0} + c_{1}a_{1} + ...c_{n}a_{n} = 0, represents that a_{0}, a_{1}... a_{n} are linearly dependent.
So rank of 'A' = 0, (so if ‘B’ rank is = 0 infinite solution, ‘B’ rank>0 no solution) ⇾(1)
Another condition given here is, 'Σa_{i} = b',
so for c_{1}c_{2}...c_{n} = {1,1,...1} set, it is having value 'b',
so there exists a solution if AX = b →(2)
From (1)&(2), we can conclude that AX = B has infinitely many solutions.
Question 9 
Let u and v be two vectors in R^{2} whose Euclidean norms satisfy u=2v. What is the value of α such that w = u + αv bisects the angle between u and v?
2  
1/2  
1  
1/2 
Let u, v be vectors in R^{2}, increasing at a point, with an angle θ.
A vector bisecting the angle should split θ into θ/2, θ/2
Means ‘w’ should have the same angle with u, v and it should be half of the angle between u and v.
Assume that the angle between u, v be 2θ (thus angle between u,w = θ and v,w = θ)
Cosθ = (u∙w)/(∥u∥ ∥w∥) ⇾(1)
Cosθ = (v∙w)/(∥v∥ ∥w∥) ⇾(2)
(1)/(2) ⇒ 1/1 = ((u∙w)/(∥u∥ ∥w∥))/((v∙w)/(∥v∥ ∥w∥)) ⇒ 1 = ((u∙w)/(∥u∥))/((v∙w)/(∥v∥))
⇒ (u∙w)/(v∙w) = (∥u∥)/(∥v∥) which is given that ∥u∥ = 2 ∥v∥
⇒ (u∙w)/(v∙w) = (2∥v∥)/(∥v∥) = 2 ⇾(3)
Given ∥u∥ = 2∥v∥
u∙v = ∥u∥ ∥v∥Cosθ
=2∙∥v∥^{2} Cosθ
w = u+αv
(u∙w)/(v∙w) = 2
(u∙(u+αv))/(v∙(u+αv)) = 2
(u∙u+α∙u∙v)/(u∙v+α∙v∙v) = 2a∙a = ∥a∥^{2}
4∥v∥^{2}+α∙2∙∥v∥^{2} Cosθ = 2(2∥v∥^{2} Cosθ+α∙∥v∥^{2})
4+2αCosθ = 2(2Cosθ+α)
4+2αCosθ = 4Cosθ+2α ⇒ Cosθ(uv)+2α4 = 0
42α = Cosθ(42α)
(42α)(Cosθ1) = 0
42α = 0
Question 10 
Consider the following statements
(i) One eigenvalue must be in [5, 5].
(ii) The eigenvalue with the largest magnitude must be strictly greater than 5.
Which of the above statements about eigenvalues of A is/are necessarily CORRECT?
Both (I) and (II)  
(I) only  
(II) only  
Neither (I) nor (II) 
be a real valued, rank = 2 matrix.
a^{2}+b^{2}+c^{2}+d^{2} = 50
Square values are of order 0, 1, 4, 9, 16, 25, 36, …
So consider (0, 0, 5, 5) then Sum of this square = 0+0+25+25=50
To get rank 2, the 2×2 matrix can be
The eigen values are,
AλI = 0 (The characteristic equation)
λ(λ)25 = 0
λ^{2}25 = 0
So, the eigen values are within [5, 5], Statement I is correct.
The eigen values with largest magnitude must be strictly greater than 5: False.
So, only Statement I is correct.
Question 11 
If the characteristic polynomial of a 3 × 3 matrix M over R (the set of real numbers) is λ^{3}  4λ^{2} + aλ + 30, a ∈ ℝ, and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is ________.
5  
6  
7  
8 
λ^{3}  4λ^{2} + aλ + 30 = 0 ⇾ (1)
One eigen value is ‘2’, so substitute it
2^{3}  4(2)^{2} + a(2) + 30 = 0
8  16 + 2a + 30 = 0
2a = 22
a = 11
Substitute in (1),
λ^{3}  4λ^{2}  11 + 30 = 0
So, (1) can be written as
(λ  2)(λ^{2}  2λ  15) = 0
(λ  2)(λ^{2}  5λ + 3λ  15) = 0
(λ  2)(λ  3)(λ  5) = 0
λ = 2, 3, 5
Max λ=5
Question 12 
Two eigenvalues of a 3 × 3 real matrix P are (2 + √1) and 3. The determinant of P is __________.
18  
15  
17  
16 
So, For the given 3×3 matrix there would be 3 eigen values.
Given eigen values are : 2+i and 3.
So the third eigen value should be 2i.
As per the theorems, the determinant of the matrix is the product of the eigen values.
So the determinant is (2+i)*(2i)*3 = 15.
Question 13 
I. If m < n, then all such systems have a solution
II. If m > n, then none of these systems has a solution
III. If m = n, then there exists a system which has a solution
Which one of the following is CORRECT?
I, II and III are true  
Only II and III are true  
Only III is true  
None of them is true 
If R(A) ≠ R(AB)
then there will be no solution.
ii) False, because if R(A) = R(AB),
then there will be solution possible.
iii) True, if R(A) = R(AB),
then there exists a solution.
Question 14 
Suppose that the eigenvalues of matrix A are 1, 2, 4. The determinant of (A^{1})^{T} is _________.
0.125  
0.126  
0.127  
0.128 
Given that eigen values are 1, 2, 4.
So, its determinant is 1*2*4 = 8
The determinant of (A^{1})^{T} = 1/ A^{T} = 1/A = 1/8 = 0.125
Question 15 
In the LU decomposition of the matrix ,if the diagonal elements of U are both 1, then the lower diagonal entry l_{22} of L is
5  
6  
7  
8 
l_{11} = 2 (1)
l_{11}u_{12} = 2
u_{12} = 2/2
u_{12} = 1  (2)
l_{21} = 4  (3)
l_{21}u_{12}+l_{22} = 9
l_{22} = 9  l_{21}u_{12} = 9  4 × 1 = 5
Question 16 
Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are –1 and 7. What are the values of a and b?
a=6, b=4  
a=4, b=6  
a=3, b=5  
a=5, b=3 
By properties,
⇒ 6=1+a and 7=a4b
⇒ a=5 ⇒ 7=54b
⇒ b=3
Question 17 
The larger of the two eigenvalues of the matrix is _________.
6  
7  
8  
9 
⇒ λ^{2}  5λ  6 = 0 ⇒ (λ  6)(λ + 1) = 0 ⇒ λ = 6, 1
∴ Larger eigen value is 6.
Question 18 
Perform the following operations on the matrix
 (i) add the third row to the second row
(ii) Subtract the third column from the first column
The determinant of the resultant matrix is ________.
0  
1  
2  
3 
Method 2: Determinant is unaltered by the operations (i) and (ii)
∴ Determinant of the resultant matrix = Determinant of the given matrix
(Since C_{1}, C_{3} are proportional i.e., C_{3} = 15C_{1})
Question 19 
In the given matrix, one of the eigenvalues is 1. the eigenvectors corresponding to the eigenvalue 1 are
⎡ 1 1 2 ⎤ ⎢ 0 1 0 ⎥ ⎣ 1 2 1 ⎦
{α(4,2,1)  α≠0, α∈R}  
{α(4,2,1)  α≠0, α∈R}  
{α(2,0,1)  α≠0, α∈R}  
{α(2,0,1)  α≠0, α∈R} 
AX = λX ⇒ (A  I)X = 0
⇒ y+2z = 0 and x+2y = 0
⇒ y = 2z and x/(2) = y
∴ x/(2) = y = 2z ⇒ x/(4) = y/2 = z/1 = α(say)
∴ Eigen vectors are {α(4,2,1  α≠0, α∈R}
Question 20 
If the following system has nontrivial solution,
 px + qy + rz = 0
qx + ry + pz = 0
rx + py + qz = 0
then which one of the following options is True?
pq+r = 0 or p = q = r  
p+qr = 0 or p = q = r  
p+q+r = 0 or p = q = r  
pq+r = 0 or p = q = r 
Question 21 
The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4by4 symmetric positive definite matrix is ________.
0  
1  
2  
3 
The finite dimensional spectral theorem says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix.
Question 22 
If the matrix A is such that
then the determinant of A is equal to
0  
1  
2  
3 
Question 23 
The product of the nonzero eigenvalues of the matrix
1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 1 0 0 0 1
is ______.
6  
7  
8  
9 
AX = λX
x_{1} + x_{5} = λx_{1}  (1)
x_{1} + x_{5} = λx_{5}  (2)
(1) + (2) ⇒ 2(x_{1} + x_{5}) = λ(x_{1} + x_{5}) ⇒ λ_{1} = 2
x_{2} + x_{3} + x_{4} = λ∙x_{2}  (4)
x_{2} + x_{3} + x_{4} = λ∙x_{3}  (5)
x_{2} + x_{3} + x_{4} = λ∙x_{4}  (6)
(4)+(5)+(6) = 3(x_{2} + x_{3} + x_{4}) = λ(x_{2} + x_{3} + x_{4} ) ⇒ λ_{2} = 3
Product = λ_{1} × λ_{2} = 2×3 = 6
Question 24 
Which one of the following statements is TRUE about every n × n matrix with only real eigenvalues?
If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.  
If the trace of the matrix is positive, all its eigenvalues are positive.  
If the determinant of the matrix is positive, all its eigenvalues are positive.  
If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive. 
• The product of the n eigenvalues of A is the same as the determinant of A. •
A: Yes, for sum to be negative there should be atleast one negative number.
B: There can be one small negative number and remaining positive, where sum is positive.
C: Product of two negative numbers is positive. So, there no need of all positive eigen values.
D: There is no need for all eigen values to be positive, as product of two negative numbers is positive.
Question 25 
Which one of the following does NOT equal to
Try to derive options from the given matrix.
Observe that col 2 + col 3 will reuse x(x+1) term
C_{2} → C_{1} + C_{2}
Question 26 
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 27 
Consider the matrix as given below.
Which one of the following provides the CORRECT values of eigenvalues of the matrix?
1, 4, 3  
3, 7, 3  
7, 3, 2  
1, 2, 3 
Question 28 
Consider the following matrix . If the eigenvalues of A are 4 and 8, then
x=4, y=10  
x=5, y=8  
x=3, y=9  
x=4, y=10 
Trace = {Sum of diagonal elements of matrix}
Here given that eigen values are 4, 8
Sum = 8 + 4 = 12
Trace = 2 + y
⇒ 2 + y = 12
y = 10
Determinant = 2y  3x
Product of eigen values = 8 × 4 = 32
2y  3x = 32
(y = 10)
20  3x = 32
12 = 3x
x = 4
∴ x = 4, y = 10
Question 29 
The following system of equations

x_{1} + x_{2} + 2x_{3} = 1
x_{1} + 2x_{2} + 3x_{3} = 2
x_{1} + 4x_{2} + ax_{3} = 4
has a unique solution. The only possible value(s) for a is/are
0  
either 0 or 1  
one of 0, 1 or 1  
any real number 
When a5 = 0, then rank(A) = rank[AB]<3,
So infinite number of solutions.
But, it is given that the given system has unique solution i.e., rank(A) = rank[AB] = 3 will be retain only if a5 ≠ 0.
Question 30 
How many of the following matrices have an eigenvalue 1?
one  
two  
three  
four 
Answer: We have only one matrix with eigen value 1.
Question 31 
Let A be a 4 x 4 matrix with eigenvalues 5, 2, 1, 4. Which of the following is an eigenvalue of
[A I] [I A]
where I is the 4 x 4 identity matrix?
5  
7  
2  
1 
(AλI)^{2}I = 0 [a^{2}b^{2} = (a+b)(ab)]
(AλI+I)(AλII) = 0
(A(λI)I)(A(λ+I)I = 0
Let us assume:
λ1=k & λ +1=k
λ =k+1 λ =k1
⇓ ⇓
for k=5; λ=4 λ =6
k=2; λ=1 λ =3
k=1; λ=2 λ = 0
k=4; λ=5 λ = 3
So, λ = 4,1,2,5,6,3,0,3
Check with the option
Option C = 2
Question 32 
Consider the set of (column) vectors defined by X = {x ∈ R^{3} x_{1}+x_{2}+x_{3} = 0, where x^{T} = [x_{1},x_{2},x_{3}]^{T}}. Which of the following is TRUE?
{[1,1,0]^{T}, [1,0,1]^{T}} is a basis for the subspace X.  
{[1,1,0]^{T}, [1,0,1]^{T}} is a linearly independent set, but it does not span X and therefore is not a basis of X.  
X is not a subspace of R^{3}  
None of the above

Question 33 
F is an n×n real matrix. b is an n×1 real vector. Suppose there are two n×1 vectors, u and v such that, u≠v and Fu=b, Fv=b. Which one of the following statements is false?
Determinant of F is zero  
There are an infinite number of solutions to Fx=b  
There is an x≠0 such that Fx=0  
F must have two identical rows 
Fu = Fv
Fu  Fv = 0
F(u  v) = 0
Given u ≠ v
F = 0 (i.e., Singular matrix, so determinant is zero)
Option A is true.
⇾ Fx = b; where F is singular
It can have no solution (or) infinitely many solutions.
Option B is true.
⇾ x ≠ 0 such that Fx = 0 is True because F is singular matrix (“stated by singular matrix rules). Option C is true.
⇾ F can two identical columns and rows.
Option D is false.
Question 34 
Consider the following system of equations in three real variables x_{l}, x_{2} and x_{3}
2x_{l}  x_{2} + 3x_{3} = 1 3x_{l} 2x_{2} + 5x_{3} = 2 x_{l} + 4x_{2} + x_{3} = 3
This system of equations has
no solution  
a unique solution  
more than one but a finite number of solutions  
an infinite number of solutions

2(2  20) +1(3 + 5) + 3(12  2)
= 44 + 8 + 30
= 6 ≠ 0
→ A ≠ 0, we have Unique Solution.
Question 35 
What are the eigenvalues of the following 2 × 2 matrix?
2 1 4 5
1 and 1  
1 and 6  
2 and 5  
4 and 1 
A = (2  λ)(5  λ)  (4) = 0
10  7λ+ λ^{2}  4 = 0
λ^{2}  7λ + 6 = 0
λ^{2}  6λ  λ + 6 = 0
(λ  6) 1(λ  6) = 0
λ = 1 (or) 6
Question 36 
The number of different n × n symmetric matrices with each element being either 0 or 1 is: (Note: power(2,x) is same as 2^{x})
power (2,n)  
power (2,n^{2})  
power (2, (n^{2} + n)/2)
 
power (2, (n^{2}  n)/2)

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 + (n1) + (n2) + ... + 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 37 
Let A, B, C, D be n × n matrices, each with nonzero determinant. If ABCD = 1, then B^{1} is
D^{1}C^{1}A^{1}  
CDA
 
ADC  
Does not necessarily exist 
ABCD = I
Pre multiply A^{1} on both sides
A^{1}ABCD = A^{1}⋅I
BCD = A^{1}
Pre multiply B^{1} on both sides
B^{1}BCD = B^{1}A^{1}
CD = B^{1}A^{1}
Post multiply A on both sides
CDA = B^{1}A^{1}⋅A
∴ CDA = B^{1}(I)
∴ CDA = B^{1}
Question 38 
How many solutions does the following system of linear equations have?
x + 5y = 1 x  y = 2 x + 3y = 3
infinitely many  
two distinct solutions  
unique  
none 
rank = r(A) = r(AB) = 2
rank = total no. of variables
Hence, unique solution.