Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 1 - Mechanical Engineering MCQ

(b) This can be written as AX = B Where A

Angemented matrix 

rank(A) ≠ rank(). The system is inconsistant .So system has no solution.

(a)

∴ rank of() = rank of(A) = 3

∴ The system has unique solution.

(b)

∴ rank (A) = rank () = 3
∴ The system has unique solution

(c )

∴ rank(A) = 2 ≠ 3 = rank() .

∴ The system is inconsistent and has no solution.

(c)
 Given  

A non-homogeneous system of linear equations representing an over-determined system (where there are more equations than unknowns) can generally be described as follows:

• If the system is consistent, it means there exists at least one solution.
• If the system is inconsistent, it means there are no solutions.

Given that the system is over-determined (more equations than unknowns), it is more likely to be inconsistent because it is harder for all equations to be satisfied simultaneously. However, if it is consistent, it would typically have a unique solution since the extra constraints (more equations) usually reduce the solution space to a single point.

Thus, the correct answer is: Inconsistent having no solution.

given set of equations are x₁+2x₂+x₃+4x₄=2 , 3x₁+6x₂+3x₃+12x₄=6

consider AB = 

⇒ R₂ → R₂ - 3R₁

AB = 

P(A)=1; P(AB)=1;n=4

⇒ P(A) =P(B) < no. of variables

⇒ Infinitely many solutions ⇒multiple non-trivial solution

(c)

(b)

Let A = 

Then C = AB = 

Then det (AB) = 0.

Then also every minor
of order 2 is also zero.
∴ rank(C) =1.

(a) (P) Singular matrix → Determinant is zero
(Q) Non-square matrix → Determinant is not defined
(R) Real symmetric → Eigen values are always real
(S) Orthogonal → Determinant is always one

(a)

(b)
(PQ) ^-1 = P Q^-1P^-1P = Q^-1

(b)

=  

Concept:

Non-homogeneous equation of type AX = B has infinite solutions;
if ρ(A | B) = ρ(A) < Number of unknowns
Calculation:

Given:

2x + 3y = 1

4x + 6y = λ

The augmented matrix is given by:

Ans. (b)

……(Lapalace formule)

Concept:

A function is odd, if the function on one side of x-axis">t-axisx-axis is sign inverted with respect to the other side or graphically, symmetric about the origin.

f(t) = - f(-t)

Symmetry condition of Fourier Transform

Calculation:

We have the wave form of function f (t) as

From the wave form, f(t) is an odd function

∴ f (t) = - f (- t)

⇒ Fourier transform of the function is imaginary and odd function of ω