Linear Algebra NAT Level - 1 - Physics MCQ

The given system will be consistent if,

The correct answer is: 3

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

The eigenvalues of M are given by :

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

The correct answer is: 4

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

The correct answer is a,b

Consider the coefficient matrix, A of the given system

Hence, rank of  A = 4 = Number of unknowns

The correct answer is: 4

⇒  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

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

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

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

The given matrix is 

Hence rank M = 2

The correct answer is: 2