Question

Find the LU decomposition of each matrix A using Doolittle's method, and then solve the systems: (a) \[ A=\left(\begin{array}{rrr} 2 & -1 & 1 \\ -3 & 4 & -1 \\ 1 & -1 & 1 \end{array}\right), \quad \ma

Answer 1

The LU decomposition of matrix A using Doolittle's method is given by:

`\[ A = \begin{bmatrix} 2 & -1 & 1 \\ -3 & 4 & -1 \\ 1 & -1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ -(3)/(2) & 1 & 0 \\ (1)/(2) & (1)/(3) & 1 \end{bmatrix} \begin{bmatrix} 2 & -1 & 1 \\ 0 & (5)/(2) & -(3)/(2) \\ 0 & 0 & (5)/(3) \end{bmatrix} \]`

Step-by-step explanation:

Doolittle's method involves decomposing matrix A into a lower triangular matrix (L) and an upper triangular matrix (U). The process starts with assuming L to be a lower triangular matrix with ones on the diagonal and U as an upper triangular matrix with elements on the diagonal. The values of L and U are then determined by Gaussian elimination.

For the given matrix A:

`\[ A = \begin{bmatrix} 2 & -1 & 1 \\ -3 & 4 & -1 \\ 1 & -1 & 1 \end{bmatrix} \]`

After applying Doolittle's method, we get:

`\[ L = \begin{bmatrix} 1 & 0 & 0 \\ -(3)/(2) & 1 & 0 \\ (1)/(2) & (1)/(3) & 1 \end{bmatrix}, \]`

`\[ U = \begin{bmatrix} 2 & -1 & 1 \\ 0 & (5)/(2) & -(3)/(2) \\ 0 & 0 & (5)/(3) \end{bmatrix} \]`

To solve a system of equations Ax = b, where b is a column matrix, we can use the LU decomposition to solve two simpler systems: Ly = b and Ux = y. By forward and backward substitution, we can find the values of x and y and ultimately the solution to the original system of equations.