Linear Algebra NAT Level - 1


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


Description
This mock test of Linear Algebra NAT Level - 1 for Physics helps you for every Physics entrance exam. This contains 10 Multiple Choice Questions for Physics Linear Algebra NAT Level - 1 (mcq) to study with solutions a complete question bank. The solved questions answers in this Linear Algebra NAT Level - 1 quiz give you a good mix of easy questions and tough questions. Physics students definitely take this Linear Algebra NAT Level - 1 exercise for a better result in the exam. You can find other Linear Algebra NAT Level - 1 extra questions, long questions & short questions for Physics on EduRev as well by searching above.
*Answer can only contain numeric values
QUESTION: 1

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


Solution:

The given system will be consistent if,


The correct answer is: 3

*Answer can only contain numeric values
QUESTION: 2

Given a matrix, 

What would be the sum of eigenvalues of A–1?


Solution:

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
QUESTION: 3

Consider the matrix  Then find the product of eigenvalues?


Solution:

The eigenvalues of M are given by :


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

The correct answer is: -4

*Answer can only contain numeric values
QUESTION: 4

The three equations,
–2x + y + z = a
x – 2y + z = b
x + y – 2z = c
has no solution. Find the value of  (a + b + c)


Solution:



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

The correct answer is: 0

*Answer can only contain numeric values
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.


Solution:

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
QUESTION: 6

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

then find the value of α.


Solution:



 

⇒  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
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


Solution:

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
QUESTION: 8

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


Solution:

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
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


Solution:

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
QUESTION: 10

 Then the rank of M is equal to :


Solution:

The given matrix is 



Hence rank M = 2

The correct answer is: 2

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