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# 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,  *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 *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.

*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.

*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

*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

*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.

*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

*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.

*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