Numerical Analysis · Linear Systems · 2 marks · Multiple choice
For Mx=b, $M=\begin{pmatrix}2&-1\\-4&3\end{pmatrix}$ and b=(−2,5)ᵀ. P:if M=LU with lower triangular L having unit diagonal, trace U=3. Q:Jacobi iterates converge to the unique solution from every initial vector. Choose truth values.
Elimination gives $L=\begin{pmatrix}1&0\\-2&1\end{pmatrix}$ and $U=\begin{pmatrix}2&-1\\0&1\end{pmatrix}$, trace 3. The Jacobi iteration matrix is $\begin{pmatrix}0&1/2\\4/3&0\end{pmatrix}$ with spectral radius √(2/3)<1, ensuring convergence from all starts. **A**.
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