https://r2.erweima.ai/mathbot/image/1726632355116trz9d8yx.jpg

We are given that matrix \( M \) is factored as \( M = LU \), where the matrices \( L \) and \( U \) are:

\[
L = \begin{pmatrix}
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 \\
1 & -1 & 0 & 1
\end{pmatrix}, 
\quad U = \begin{pmatrix}
1 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 2 \\
0 & 0 & 0 & 2
\end{pmatrix}
\]

We are also given that the vector \( b \) is:

\[
b = \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix}
\]

We are asked to solve the system \( Mx = b \), and find the first coordinate of the vector \( x \).

### Solution Strategy
We solve the system using the LU decomposition \( M = LU \), which gives:

\[
LUx = b
\]

This can be broken into two steps:
1. Solve \( Ly = b \) for \( y \),
2. Solve \( Ux = y \) for \( x \).

### Step 1: Solve \( Ly = b \)

The matrix \( L \) is lower triangular, so we can use forward substitution to solve for \( y \):

\[
L = \begin{pmatrix}
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 \\
1 & -1 & 0 & 1
\end{pmatrix}, 
\quad y = \begin{pmatrix} y_1 \\ y_2 \\ y_3 \\ y_4 \end{pmatrix}, 
\quad b = \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix}
\]

From the matrix equation \( Ly = b \), we get the following system of equations:

1. \( y_1 = 1 \),
2. \( y_1 + y_2 = 0 \) \( \Rightarrow y_2 = -1 \),
3. \( y_1 + y_2 = 1 \) \( \Rightarrow 1 - 1 = 1 \), which is inconsistent.

It appears there is an error in the setup of the problem. Let me recompute and

A matrix M can be factored as M = LU, where L = [1 0 0 0; 1 1 0 0; 1 1 0 0; 1 -1 0 1] and U = [1 0 1 0; 0 1 1 0; 0 0 1 2; 0 0 0 2]. If b = [1; 0; 1; 0] and x is the solution of the system Mx = b, what is the first coordinate of the vector x?

Learn how to solve a system of linear equations using LU decomposition where M = LU. This example involves solving for x in Mx = b with matrices L and U provided, using forward and backward substitution techniques.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation