Linear Algebra NAT Level - 1 - Physics MCQ

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,

*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

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*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.

*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.

*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

*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

*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.

*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

*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.

*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

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