Question 1 |
The number of distinct eigenvalues of the matrix 
is equal to __________.
is equal to __________. Fill in the Blank Type Question |
Question 1 Explanation:
Question 2 |
Consider the 5 × 5 matrix 
It is given that A has only one real Eigen value. Then the real Eigen value of A is
It is given that A has only one real Eigen value. Then the real Eigen value of A is -2.5 | |
0 | |
15 | |
25 |
Question 2 Explanation:
If sum of all the rows or columns are same, then that
sum = one eigen value of that matrix
Sum of all rows
(15-x) * |Matrix| = 0
15 is a factor .
Another Approach
sum = one eigen value of that matrix
Sum of all rows
(15-x) * |Matrix| = 0
15 is a factor .
Another Approach
Question 3 |
The rank of the matrix M =
is
is 0 | |
1 | |
2 | |
3 |
Question 3 Explanation:
|M |= 
= 5(0-12) – 10(6-6) +10(6-0)= -60-0+60=0
But a 2×2 minor,
=0-10=-10≠0
Rank = 2
= 5(0-12) – 10(6-6) +10(6-0)= -60-0+60=0
But a 2×2 minor,
=0-10=-10≠0 Rank = 2Question 4 |
Consider the following statement about the linear dependence of the real valued functions y1= 1, y2= x and y 3 =x2, over the field of real numbers.
I. y1, y2 and y3 are linearly independent on -1 ≤ x ≤ 0
II. y1 , y2 and y3 are linearly dependent on 0 ≤ x ≤1
III. y1 , y2 and y3 are linearly independent on 0 ≤x ≤1
IV. y1, y2 and y3 are linearly dependent on -1 ≤ x ≤ 0
Which one among the following is correct?
Which one among the following is correct? | |
Both I and III are true | |
Both II and IV are true | |
Both III and IV are true |
Question 4 Explanation:
Given
y1= 1, y 2= x, y 3= x2
Consider
=
=2 ≠0
y1, y2, y3 are linearly independent 
x
y1= 1, y 2= x, y 3= x2
Consider
=
=2 ≠0 y1, y2, y3 are linearly independent x Question 5 |
Consider matrix 
and vector 
The number of distinct real values of 
for which the equation 
has infinitely many solution is ___________.
and vector The number of distinct real values of for which the equation has infinitely many solution is ___________. Fill in the Blank Type Question |
Question 5 Explanation:
Question 6 |
Let M4 = I, (where I denotes the identity matrix) and M ≠ I, M2 ≠I and M3 ≠I. Then, for any natural number k, M-1 equals:
M4k+1 | |
M4k + 2 | |
M4k+3 | |
M4k |
Question 6 Explanation:
Question 7 |
The matrix 
has det A. = 100 and trace A=14
The value of
is
has det A. = 100 and trace A=14 The value of
is Fill in the Blank Type Question |
Question 7 Explanation:
Question 8 |
Consider a 
square matrix
Where x is unknown. If the eigenvalues of the matrix A are
and 
the x is equal to
square matrix Where x is unknown. If the eigenvalues of the matrix A are and the x is equal to  | |
 | |
 | |
 |
Question 8 Explanation:
Given
Product of eigen values = Det of A
Product of eigen values = Det of A
Question 9 |
For A = 
, the determinant of AT A–1 is
, the determinant of AT A–1 is sec2 x | |
cos 4x | |
1 | |
0 |
Question 9 Explanation:
Question 10 |
For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold?
 | |
(cM)T = c(M)T | |
 | |
 |
Question 10 Explanation:
Matrix multiplication is not commutative in general.
All other are properties of matrix.
Refer: www.khanacademy.org
All other are properties of matrix.
Refer: www.khanacademy.org
Question 11 |
A real (4 × 4) matrix A satisfies the equation A2 = I, where I is the (4 × 4) identity matrix. The positive eigen value of A is __________.
1 | |
2 | |
4 | |
8 |
Question 11 Explanation:
Question 12 |
Consider the matrix:
Which is obtained by reversing the order of the columns of the identity matrix I6.
Let P= I6 + αJ6 where α is a non-negative real number.
The value of α for which det(P) = 0 is ___________.
Which is obtained by reversing the order of the columns of the identity matrix I6.
Let P= I6 + αJ6 where α is a non-negative real number.
The value of α for which det(P) = 0 is ___________.
0 | |
1 | |
2 | |
4 |
Question 12 Explanation:
Since, α is non-negative real number. Hence,
α = 1
Question 13 |
The determinant of matrix A is 5 and the determinant of matrix B is 40. The determinant of matrix AB is ________.
100 | |
8 | |
1000 | |
200 |
Question 13 Explanation:
Determinant of matrix AB will be
Determinant of (A) * Determinant of (B)
5*40 = 200
so answer = 200
Determinant of (A) * Determinant of (B)
5*40 = 200
so answer = 200
Question 14 |
The system of linear equations
a unique solution | |
infinitely many solutions | |
no solution | |
exactly two solutions |
Question 14 Explanation:
Clearly rank(A)=2 , rank(A/B) = 2 , number of unknowns = 3
so rank (A) = rank (A / B) =2
Since, rank (A) = rank (A / B) < number of unknowns
∴ Equations have infinitely many solutions.
Question 15 |
The maximum value of the determinant among all 2×2 real symmetric matrices with trace 14 is ________.
40 | |
45 | |
49 | |
52 |
Question 15 Explanation:
Question 16 |
Which one of the following statements is NOT true for a square matrix?
If A is upper triangular, the eigenvalues of A are the diagonal elements of it | |
If A is real symmetric, the eigenvalues of A are always real and positive | |
If A is real, the eigenvalues of A and AT are always the same | |
If all the principal minors of A are positive, all the eigenvalues of A are also positive |
Question 16 Explanation:
Question 17 |
Let A be an m × n matric and B an n × m matric. It is given that determinant 
determinant 
where 
is the k × k identity matrix. Using the above property, the determinant of the matrix given below is
determinant where is the k × k identity matrix. Using the above property, the determinant of the matrix given below is 2 | |
5 | |
8 | |
16 |
Question 17 Explanation:
Question 18 |
Given that 
and 
, the value of A3 is
and , the value of A3 is 15 A + 12 I | |
19A + 30 I | |
17A + 15 I | |
17 A + 21I |
Question 18 Explanation:
Characteristic equation of A is
(-5-λ)(-λ) + 6 = 0
So,
(by Cayley Hamilton theorem)
⇒ A2 = –5A – 6I
Multiplying by A on both sides, we have,
A3 =-5A2-6A
⇒ A3 = -5(-5A-6I)-6A
= 19A + 30I
(-5-λ)(-λ) + 6 = 0
So,
(by Cayley Hamilton theorem)
⇒ A2 = –5A – 6I
Multiplying by A on both sides, we have,
A3 =-5A2-6A
⇒ A3 = -5(-5A-6I)-6A
= 19A + 30I
There are 18 questions to complete.
I cant solve the questioner! because its so hard!
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right answer of Question no. 9 is 1
no, its correct
Yes
Answer for question no 9 is option C that is 1. A inverse can be written an 1/A and A transpose is same as A ….so A*(1/A) gets cancelled and the answer is 1.
Answer for question no 9 is option C that is 1. A inverse can be written an 1/A and A transpose is same as A ….so A*(1/A) gets cancelled and the answer is 1.
A transpose is not same as matrix A