Math Problem Statement
solve above question step by step
Solution
Here is the step-by-step solution to the system using LU factorization:
-
Matrix is factorized into two matrices: (lower triangular) and (upper triangular) using LU decomposition:
- ( L = \begin{pmatrix} 1 & 0 & 0 & 0 \ 2 & 1 & 0 & 0 \ -1 & 0 & 1 & 0 \ -4 & 3 & -5 & 1 \end{pmatrix} )
- ( U = \begin{pmatrix} 1 & -2 & -4 & -3 \ 0 & -3 & 1 & 0 \ 0 & 0 & 2 & 1 \ 0 & 0 & 0 & 1 \end{pmatrix} )
-
Solving for in the equation : We solve the lower triangular system, and the solution is:
-2 \\ -1 \\ 2 \\ -3 \end{pmatrix}$$ -
Solving for in the equation : We solve the upper triangular system, and the solution to the system is:
-2 \\ -1 \\ 2 \\ -3 \end{pmatrix}$$
Thus, the solution to the system is .
Would you like more details or clarifications on any step?
Here are some related questions to explore:
- How does LU factorization work in general?
- What are the differences between LU and QR factorization?
- Can LU factorization always be applied to any matrix?
- How would the solution change if were singular?
- What are the practical applications of LU factorization?
Tip: LU factorization is particularly useful when solving multiple systems with the same matrix but different vectors .
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Factorization
Formulas
LU Decomposition: A = LU
Solving LY = b
Solving UX = Y
Theorems
LU Factorization
Suitable Grade Level
Undergraduate Level
Related Recommendation
LU Factorization and Solving Ax = b Using Matrix Decomposition
Solve System of Linear Equations Using LU Factorization
LU Factorization of a 3x4 Matrix Using Gaussian Elimination
Solving a 4x4 System of Equations using Gaussian Elimination and Matrix Inversion
LU Decomposition and Solving LUX = b for a Given Matrix