MTH603 Grand Quiz Solution and Discussion

@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.


Whileusingtherelaxationmethodforfindingthesolutionofthefollowingsystem11x1+x2−x3=8 x1+8x2+5x3=9 x1+x2+9x3=7withtheinitialvector(0,0,0),theresidualswouldbe

By using determinants, we can easily check that the solution of the given system of linear equation ______ and it is ______.

If the determinant of a matrix A is not equal to zero then the system of equations will have……….
A nxn nonhomogeneous system of linear equations has a unique nontrivial solution if and only if its determinant is nonzero. If this determinant is zero, then the system has either no nontrivial solutions or an infinite number of solutions.

For the system of equations; x =2, y=3. The inverse of the matrix associated with its coefficients is.