MTH603 Grand Quiz Solution and Discussion

@zaasmi said in MTH603 Grand Quiz Solution and Discussion:
While solving a system of linear equations by Gauss Jordon Method, after all the elementary row operations if there lefts also zeros on the main diagonal then which of the is true about the system?

While solving a system of linear equations by Gauss Jordon Method, after all the elementary row operations if there lefts also zeros on the main diagonal then which of the is true about the system?

@zaasmi said in MTH603 Grand Quiz Solution and Discussion:
Eigenvectors of a symmetric matrix are orthogonal, but only for distinct eigenvalues.
The eigenvectors of a symmetric matrix A corresponding to different eigenvalues are orthogonal to each other.

Eigenvectors of a symmetric matrix are orthogonal, but only for distinct eigenvalues.

@zaasmi said in MTH603 Grand Quiz Solution and Discussion:
If n x n matrices A and B are similar, then they have the different eigenvalues (with the same multiplicities).
Since similar matrices A and B have the same characteristic polynomial, they also have the same eigenvalues. If B = PAP−1 and v = 0 is an eigenvector of A (say Av = λv) then B(Pv) = PAP−1(Pv) = PA(P−1P)v = PAv = λPv. Thus Pv (which is nonzero since P is invertible) is an eigenvector for B with eigenvalue λ.

If n x n matrices A and B are similar, then they have the different eigenvalues (with the same multiplicities).

@zaasmi said in MTH603 Grand Quiz Solution and Discussion:
Eigenvalues of a _________ matrix are all real.
If each entry of an n×n matrix A is a real number, then the eigenvalues of A are all real numbers. False. In general, a real matrix can have a complex number eigenvalue. In fact, the part (b) gives an example of such a matrix.

Eigenvalues of a _________ matrix are all real.

@zaasmi said in MTH603 Grand Quiz Solution and Discussion:
The Jacobi iteration converges, if A is strictly diagonally dominant.
If A is strictly row diagonally dominant, then the Jacobi iteration converges for any choice of the initial approximation x(0). However, the Jacobi iteration may converge for a matrix that is not strictly row diagonally dominant.

The Jacobi iteration converges, if A is strictly diagonally dominant.

The linear equation 2x+0y2=0 has solution(s).


Let[A]bea3×3realsymmetricmatrixwitha23bethenumericallylargestoff−diagonalelementthenusingJacobi′smethodthevalueofθcanbefoundby
Let A be a 3×3 matrix with real entries. Prove that if A is not similar over
R to a triangular matrix then A is similar over C to a diagonal matrix. 
While using Relaxation method, which of the following is increment ‘dxi’corresponding to the largest Residual for 1st iteration on the system;
2x+3y = 1, 3x +2y =  4 ? 
Central Difference method is the finite difference method
Finite difference. A finite difference is a mathematical expression of the form f (x + b) − f (x + a). … The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
