dy/dx - = 1 - y,y(0) = 0 is an example of

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

A 3 x 3 identity matrix have three and __________eigen values.

same
different

A (3 \times 3) identity matrix has three eigenvalues, and they are all the same.

Explanation:

• An identity matrix (I) of size (n \times n) has all its diagonal elements equal to 1 and all off-diagonal elements equal to 0.

• The eigenvalues of an (n \times n) identity matrix are all equal to 1.

For a (3 \times 3) identity matrix, the eigenvalues are (1, 1, 1), meaning they are all the same.

So the correct answer is:

same

[zaasmi]

Eigenvalues of a symmetric matrix are all _______ .
Real
complex
zero
positive

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Eigenvalues of a symmetric matrix are all _______ .
Real
complex
zero
positive

The eigenvalues of a symmetric matrix are all:

Real

Explanation:

• Symmetric Matrix: A matrix (A) is symmetric if (A = A^T) (i.e., (A) is equal to its transpose).

• The eigenvalues of a symmetric matrix are always real numbers, regardless of whether the matrix is positive definite or not.

• Symmetric matrices may also have complex eigenvalues, but only in the case of non-real matrices. For symmetric matrices, the eigenvalues are guaranteed to be real.

So the correct option is:

Real

[zaasmi]

The Jacobi iteration converges, if A is strictly diagonally dominant.
TRUE (Page 69)
FALSE

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

The Jacobi iteration converges, if A is strictly diagonally dominant.
TRUE (Page 69)
FALSE

TRUE

Explanation:

• Jacobi Iteration: This is an iterative method used to solve a system of linear equations. It involves iterating on the solution vector using the diagonal elements of the coefficient matrix.

• Strict Diagonal Dominance: A matrix is strictly diagonally dominant if, for each row of the matrix, the magnitude of the diagonal element is greater than the sum of the magnitudes of all other (non-diagonal) elements in that row.

• Convergence: The Jacobi iteration method converges if the coefficient matrix ( A ) is strictly diagonally dominant. This means that the method will approach the correct solution if this condition is met.

So the correct statement is:

TRUE

[zaasmi]

Below are all the finite difference methods EXCEPT _________.
jacobi’s method
newton’s backward difference method
Stirlling formula
Forward difference method

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Below are all the finite difference methods EXCEPT _________.
jacobi’s method
newton’s backward difference method
Stirlling formula
Forward difference method

The correct answer is:

Jacobi’s method

Explanation:

• Jacobi’s Method: This is an iterative method for solving a system of linear equations, not a finite difference method.

• Newton’s Backward Difference Method: This is a finite difference method used for interpolation and numerical differentiation.

• Stirling Formula: This is used for interpolation in finite difference methods and approximates the values of the function using finite differences.

• Forward Difference Method: This is used in finite difference methods to approximate derivatives and solve differential equations.

Thus, Jacobi’s method is not a finite difference method, so it is the correct choice for the given question.

[zaasmi]

Two matrices with the same characteristic polynomial need not be similar.
TRUE
FALSE

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Two matrices with the same characteristic polynomial need not be similar.
TRUE
FALSE

TRUE

Explanation:

• Similar Matrices: Two matrices (A) and (B) are similar if there exists an invertible matrix (P) such that (B = P^{-1}AP). Similar matrices share the same eigenvalues and their Jordan forms are the same, which means they have the same characteristic polynomial.

• Characteristic Polynomial: Matrices that have the same characteristic polynomial are guaranteed to have the same eigenvalues, but this alone does not guarantee similarity. Similar matrices must also have the same Jordan canonical form or must be related by a similarity transformation.

Therefore, having the same characteristic polynomial is a necessary but not sufficient condition for similarity.

So, the correct statement is:

TRUE

[zaasmi]

The determinant of a diagonal matrix is the product of the diagonal elements.
True
False

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

The determinant of a diagonal matrix is the product of the diagonal elements.
True
False

True

Explanation:

• For a diagonal matrix, all off-diagonal elements are zero. The determinant of a diagonal matrix is calculated as the product of its diagonal elements.

For example, if (D) is a diagonal matrix with diagonal elements (d_1, d_2, \ldots, d_n), then:

[ \text{det}(D) = d_1 \cdot d_2 \cdot \ldots \cdot d_n ]

So, the determinant of a diagonal matrix is indeed the product of its diagonal elements.

Therefore, the statement is:

True

[zaasmi]

The Gauss-Seidel method is applicable to strictly diagonally dominant or symmetric positive definite
matrices A.
True
False

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

The Gauss-Seidel method is applicable to strictly diagonally dominant or symmetric positive definite
matrices A.
True
False

True

Explanation:

• Gauss-Seidel Method: This iterative method for solving linear systems converges under certain conditions.

• Strictly Diagonally Dominant Matrices: If a matrix is strictly diagonally dominant (i.e., for each row of the matrix, the magnitude of the diagonal element is greater than the sum of the magnitudes of the other elements in that row), the Gauss-Seidel method will converge.

• Symmetric Positive Definite Matrices: For symmetric positive definite matrices, the Gauss-Seidel method is guaranteed to converge.

Therefore, the Gauss-Seidel method is indeed applicable and convergent for strictly diagonally dominant or symmetric positive definite matrices.

So, the correct statement is:

True

[zaasmi]

The determinant of a _______ matrix is the product of the diagonal elements.

1. Upper triangular
2. Lower triangular
3. Scalar
4. Diagonal

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

The determinant of a _______ matrix is the product of the diagonal elements.

1. Upper triangular
2. Lower triangular
3. Scalar
4. Diagonal

The determinant of a matrix is the product of the diagonal elements if the matrix is:

Diagonal, Upper Triangular, or Lower Triangular

Explanation:

• Diagonal Matrix: All off-diagonal elements are zero. The determinant is the product of the diagonal elements.

• Upper Triangular Matrix: All elements below the main diagonal are zero. The determinant is the product of the diagonal elements.

• Lower Triangular Matrix: All elements above the main diagonal are zero. The determinant is the product of the diagonal elements.

• Scalar Matrix: A diagonal matrix where all diagonal elements are the same scalar value. The determinant is still the product of the diagonal elements.

Therefore, the correct options are:

Diagonal, Upper Triangular, Lower Triangular

[zaasmi]

For differences methods we require the set of values.
True
False