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Linear Algebra NAT Level - 1 - Physics MCQ


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10 Questions MCQ Test Topic wise Tests for IIT JAM Physics - Linear Algebra NAT Level - 1

Linear Algebra NAT Level - 1 for Physics 2024 is part of Topic wise Tests for IIT JAM Physics preparation. The Linear Algebra NAT Level - 1 questions and answers have been prepared according to the Physics exam syllabus.The Linear Algebra NAT Level - 1 MCQs are made for Physics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Linear Algebra NAT Level - 1 below.
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*Answer can only contain numeric values
Linear Algebra NAT Level - 1 - Question 1

What is the maximum value of λ, for which the given system is inconsistent?


Detailed Solution for Linear Algebra NAT Level - 1 - Question 1

The given system will be consistent if,


The correct answer is: 3

*Answer can only contain numeric values
Linear Algebra NAT Level - 1 - Question 2

Given a matrix, 

What would be the sum of eigenvalues of A–1? (Give the answer upto 3 decimal places ) 


Detailed Solution for Linear Algebra NAT Level - 1 - Question 2

The characteristic equation of A is,

∴   the characteristic root of A are given by

λ = 2, 2, 8

We know that if  are the eigenvalues of  A, then  will be the eigenvalues of A–1.

Hence, in this case, eigenvalues of  A–1 will be 

∴  Sum of eigenvalues of 

The correct answer is: 1.125

*Answer can only contain numeric values
Linear Algebra NAT Level - 1 - Question 3

Consider the matrix  Then find the product of eigenvalues?


Detailed Solution for Linear Algebra NAT Level - 1 - Question 3

The eigenvalues of M are given by :


⇒ Product of eigenvalues = 4
So, M  both positive and negative real eigenvalues.

The correct answer is: 4

*Multiple options can be correct
Linear Algebra NAT Level - 1 - Question 4

The three equations,
–2x + y + z = a
x – 2y + z = b
x + y – 2z = c
has 

Detailed Solution for Linear Algebra NAT Level - 1 - Question 4



Hence, the system won't contain any solution unless a + b + c becomes 0.

The correct answer is a,b

*Answer can only contain numeric values
Linear Algebra NAT Level - 1 - Question 5

Consider the following equations
x + y – 3z + 2w = 0
2x – y + 2z – 3w = 0
3x – 2y + z – 4w = 0
–4x – y + 3z + 4w = 0

Find the rank of the coefficient matrix.


Detailed Solution for Linear Algebra NAT Level - 1 - Question 5

Consider the coefficient matrix, A of the given system



Hence, rank of  A = 4 = Number of unknowns

The correct answer is: 4

*Answer can only contain numeric values
Linear Algebra NAT Level - 1 - Question 6

If x = y = z = αc satisfy the matrix equation AX = 0 where

then find the value of α.


Detailed Solution for Linear Algebra NAT Level - 1 - Question 6



 

⇒  Rank of A = 2 and z  is a free variable. The given system reduces to
x + 2y + 3z = 0
y – 2z = 0
Let  z = 0
⇒ y = 0
and  x = –2y – 3z = 0
Hence,  x = y = z = 0  gives the general solution.
⇒  α = 0

The correct answer is: 0

*Answer can only contain numeric values
Linear Algebra NAT Level - 1 - Question 7

For which value of λ will the following equations fail to have unique solution?

3x – y + λz = 1 
2x + y + z = 2
x + 2y – λz = –1


Detailed Solution for Linear Algebra NAT Level - 1 - Question 7

The matrix form of the given system of equations is


The given system of equations will have a unique solution if and only if the coefficient matrix is non singular.

Performing  we get

Performing  we get

Therefore, the coefficient matrix will be non-singular if and only if

i.e., if and only if 

Thus, the given system will have a unique solution if  In case of  the equation (1) becomes

Performing  we get


showing that given equations are inconsistent in this case.

Thus if  no solution exists.

The correct answer is: -3.5

*Answer can only contain numeric values
Linear Algebra NAT Level - 1 - Question 8

Let A be a 3 × 3 matrix with eigen values 1, –1, 0. Then the determinant of I + A100 is :


Detailed Solution for Linear Algebra NAT Level - 1 - Question 8

Eigen values of 

⇒ Eigen values of 
Eigen values of = 1, 1, 0
Eigen values of 

⇒  Eigen values of 
Hence,  |A100 + I| = 2 × 2 × 1 = 4

The correct answer is: 4

*Answer can only contain numeric values
Linear Algebra NAT Level - 1 - Question 9

The system of equations αx + y + z = α - 1, x + αy + z = α - 1, x + y + αz = α - 1 has no solution. Find the maximum value of α


Detailed Solution for Linear Algebra NAT Level - 1 - Question 9

The given system is,



The system won't have any solution if rank (A) ≠ Rank(A : B) i.e.

∴   α = either 2 or –1 but not equals to 0 or 3.

The correct answer is: 2

*Answer can only contain numeric values
Linear Algebra NAT Level - 1 - Question 10

 Then the rank of M is equal to :


Detailed Solution for Linear Algebra NAT Level - 1 - Question 10

The given matrix is 



Hence rank M = 2

The correct answer is: 2

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