Question

Given matrix A = [6 -2 0; 9 -1 1; 3 7 5], find the LU decomposition of A [you may use a calculator to find L and U matrices].

Answer 1

The LU decomposition of the given matrix involves finding two matrices, L and U, such that A = LU. To find the L and U matrices, perform row operations to eliminate the entries below the diagonal in the first column and keep track of the row operations to determine the entries in L. The resulting matrices are L = [1 0 0; -9/6 1 0; -3/6 0 1] and U = [6 -2 0; 0 5/2 1; 0 0 4/3].

Step-by-step explanation:

The LU decomposition of a matrix A involves finding two matrices, L and U, such that A = LU. Here is the step-by-step process to find the LU decomposition of the given matrix:

1. Start with the given matrix A = [6 -2 0; 9 -1 1; 3 7 5].
2. Perform row operations to eliminate the entries below the diagonal in the first column:

• Multiply the first row by -9/6 and add it to the second row.
• Multiply the first row by -3/6 and add it to the third row.

The resulting matrix will be the upper triangular matrix U.To find the lower triangular matrix L, we can keep track of the row operations we performed:

• The first row operation was adding -9/6 times the first row to the second row, so the second row of L would be [0 -9/6 0].
• The second row operation was adding -3/6 times the first row to the third row, so the third row of L would be [0 0 -3/6].

Therefore, the LU decomposition of matrix A is: A = LU, whereL = [1 0 0; -9/6 1 0; -3/6 0 1]U = [6 -2 0; 0 5/2 1; 0 0 4/3]