Systems of linear equations - Numerical Analysis - Mathematics Previous Year Questions

Consider the system of equations \[\begin{bmatrix} 5 & 2 & 1 \\ -2 & 5 & 2 \\ -1 & 2 & 8 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 13 \\ -22 \\ 14 \end{bmatrix}\] With the initial guess of the solution \([x_1^{(0)}, x_2^{(0)}, x_3^{(0)}]^T = [1, 1, 1]^T\), the approximate value of the solution \([x_1^{(1)}, x_2^{(1)}, x_3^{(1)}]^T\) after one iteration by the Gauss-Seidel method is

Let \( J \) be the Jacobi iteration matrix of the linear system \( \begin{bmatrix} 1 & 2 & 1 \\ 2 & 1 & 2 \\ -4 & 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \).
Consider the following statements:
(P): One of the eigenvalues of \( J \) lies in the interval \( [2,3] \).
(Q): The Jacobi iteration converges for the above system.
Which of the above statements hold TRUE?

Consider the linear system of equations \(Ax = b\) with \[ \[A = \begin{pmatrix} 3 & 1 & 1 \\ 1 & 4 & 1 \\ 2 & 0 & 3 \end{pmatrix} \quad \text{and} \quad b = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}.\] \] Which of the following statements are TRUE?

Consider the linear system \(Mx = b\), where \(M = \begin{bmatrix} 2 & -1 \\ -4 & 3 \end{bmatrix}\) and \(b = \begin{bmatrix} -2 \\ 5 \end{bmatrix}\).
Suppose \(M = LU\), where \(L\) and \(U\) are lower triangular and upper triangular square matrices, respectively. Consider the following statements:
\(P:\) If each element of the main diagonal of \(L\) is 1, then \(trace(U) = 3\).
\(Q:\) For any choice of the initial vector \(x^{(0)}\), the Jacobi iterates \(x^{(k)}, k = 1,2,3 ...\) converge to the unique solution of the linear system \(Mx = b\).
Then

Let
\[A = \begin{pmatrix} 2 & -1 & 1 \\ 1 & 2 & -1 \\ -1 & 1 & 2 \end{pmatrix}\]
and \(\mathbf{b}\) be a \(3 \times 1\) real column vector. Consider the statements.
I.    The Jacobi iteration method for the system \((A + \varepsilon I_3)\mathbf{x} = \mathbf{b}\) converges for any initial approximation and \(\varepsilon > 0\).
II.    The Gauss–Seidel iteration method for the system \((A + \varepsilon I_3)\mathbf{x} = \mathbf{b}\) converges for any initial approximation and \(\varepsilon > 0\).
Which one of the following is correct?

Consider the linear system \(A\mathbf{x} = \mathbf{b}\), where \(A = [a_{ij}]\), \(i, j = 1, 2, 3\), and \(a_{ii} \neq 0\) for \(i = 1, 2, 3\), is a matrix with entries in \(\mathbb{R}\). For \(D = \begin{bmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{bmatrix}\), let \(D^{-1}A = \begin{bmatrix} 1 & 1 & -2 \\ 3 & 1 & 2 \\ 1 & 1 & 1 \end{bmatrix}\) and \(D^{-1}\mathbf{b} = \begin{bmatrix} 4 \\ 4 \\ 1 \end{bmatrix}\). Consider the following two statements: S1: The approximation of \(\mathbf{x}\) after one iteration of the Jacobi scheme with initial vector \(\mathbf{x}_0 = [1 \, 1 \, 1]^T\) is \(\mathbf{x}_1 = [5 \, -1 \, -1]^T\). S2: There exists an initial vector \(\mathbf{x}_0\) for which Jacobi iterative scheme diverges. Then, which one of the following is correct?

Consider the system of linear equations \[\begin{aligned} x + y + z &= 3 \\ x - y - z &= 4 \\ x - 5y + kz &= 6. \end{aligned}\] Then the value of \(k\) for which this system has an infinite number of solutions is

If Jacobi method is used to solve the following system of linear equations \(\begin{pmatrix} 1 & 2 & 1 \\ 0 & 2 & 2 \\ 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}\) with the initial guess \(x^{(0)} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\) and \(x^{(i)} = \begin{pmatrix} x_1^{(i)} \\ x_2^{(i)} \\ x_3^{(i)} \end{pmatrix}, i = 1, 2, 3, \ldots\), denotes the \(i^{th}\) iterate, then the value of \(|x_1^{(2)} + x_2^{(2)} + x_3^{(2)}|\) is equal to ______. (answer in integer)