Question

Consider the linear system \(Mx = b\), where \(M = \begin{pmatrix} 2 & -1 \\ -4 & 3 \end{pmatrix}\) and \(b = \begin{pmatrix} -2 \\ 5 \end{pmatrix}\).
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 \(\text{trace}(U) = 3\).
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

• A. both P and Q are TRUE
• B. P is FALSE and Q is TRUE
• C. P is TRUE and Q is FALSE
• D. both P and Q are FALSE

Hint:

For LU factorization, remember that \(\det(M) = \det(L)\det(U)\). If L is unit triangular, \(\det(L)=1\), so \(\det(M)=\det(U)\). For Jacobi convergence, the sufficient condition is strict diagonal dominance. If that fails, you must check the necessary and sufficient condition: \(\rho(T_J)<1\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question has two independent parts. Part P deals with the properties of LU decomposition (specifically the Doolittle factorization). Part Q deals with the convergence condition for the Jacobi iterative method.
Step 2: Key Formula or Approach:
For P: Perform LU decomposition with \(l_{ii}=1\). The trace of U is the sum of its diagonal elements. Since L is unit lower triangular, \(\det(L)=1\), so \(\det(M)=\det(U)\). For a triangular matrix, the determinant is the product of diagonal elements.
For Q: The Jacobi method converges for any initial vector if and only if the iteration matrix \(T_J = -D^{-1}(L+U)\) has spectral radius \(\rho(T_J)<1\). A sufficient condition is that M is strictly diagonally dominant.
Step 3: Detailed Calculation:
Analysis of Statement P:
Given \(M = \begin{pmatrix} 2 & -1 \\ -4 & 3 \end{pmatrix}\) and \(L = \begin{pmatrix} 1 & 0 \\ l_{21} & 1 \end{pmatrix}\), \(U = \begin{pmatrix} u_{11} & u_{12} \\ 0 & u_{22} \end{pmatrix}\):
\[ LU = \begin{pmatrix} u_{11} & u_{12} \\ l_{21}u_{11} & l_{21}u_{12} + u_{22} \end{pmatrix} \] Comparing with M:
1. \(u_{11} = 2\)
2. \(u_{12} = -1\)
3. \(l_{21}u_{11} = -4 \implies l_{21} = -2\)
4. \(l_{21}u_{12} + u_{22} = 3 \implies 2 + u_{22} = 3 \implies u_{22} = 1\)
Thus, \(U = \begin{pmatrix} 2 & -1 \\ 0 & 1 \end{pmatrix}\) and \(\text{trace}(U) = 2 + 1 = 3\).
P is TRUE.
Analysis of Statement Q:
Jacobi iteration matrix: \(T_J = -D^{-1}(L+U)\).
\[ D = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix},\quad L+U = \begin{pmatrix} 0 & -1 \\ -4 & 0 \end{pmatrix},\quad D^{-1} = \begin{pmatrix} 1/2 & 0 \\ 0 & 1/3 \end{pmatrix} \] \[ T_J = - D^{-1}(L+U) = \begin{pmatrix} 0 & 1/2 \\ 4/3 & 0 \end{pmatrix} \] Eigenvalues of \(T_J\): \(\lambda = \pm \sqrt{(1/2)(4/3)} = \pm \sqrt{2/3}\).
Since \(\sqrt{2/3} < 1\), the Jacobi method converges for any initial vector.
Q is TRUE.
Step 4: Final Answer:
Both P and Q are TRUE.
Step 5: Why This is Correct:
P was verified by explicit LU decomposition and trace computation. Q was verified by computing the Jacobi iteration matrix, finding its eigenvalues, and confirming the spectral radius is less than 1, ensuring convergence.