# MTH603 Grand Quiz Solution and Discussion

• Power method is applicable if the eigen values are ______________.

The Jacobi’s method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal. Power method is applicable if the eigen vectors corresponding to eigen values are linearly independent.

The Power Method is used to find a dominant eigenvalue (one having the largest absolute value), if one exists, and a corresponding eigenvector. To apply the Power Method to a square matrix A, begin with an initial guess u0 for the eigenvector of the dominant eigenvalue.

• Power method is applicable if the eigen values are ______________.

• 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?

• 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.

• 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 non-zero 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).

• 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.

• 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+0y-2=0 has----- solution(s).

• Let[A]bea3×3realsymmetricmatrixwith|a23|bethenumericallylargestoff−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 ?

3

4

1

2

7

1

33

5
| |