Final answer:
The LU factorization of matrix A = [1 2 3, 2 5 6, 1 3 4] results in an L matrix of [1 0 0, 2 1 0, 1 1 1] and a U matrix of [1 2 3, 0 -1 -6, 0 0 -3]. This process involves using elementary row operations to form a lower triangular matrix L and an upper triangular matrix U.
Step-by-step explanation:
The LU factorization of a matrix involves decomposing the given matrix into the product of a lower triangular matrix (L) and an upper triangular matrix (U). To find the LU factorization of the matrix A = [1 2 3, 2 5 6, 1 3 4], we must perform a series of elementary row operations to transform A into an upper triangular matrix, U. Simultaneously, we record the row operations in the form of a lower triangular matrix, L.
Let's proceed with the factorization:
- Starting with the original matrix A, we notice that the first entry is already 1, which is our pivot.
- To zero out the entries below the pivot in the first column, we can perform the following row operations: R2 → R2 - 2 * R1 and R3 → R3 - R1.
- After performing these operations, the obtained upper triangular matrix is U and the corresponding lower triangular matrix is L with the elements reflecting the row operations we performed.
The complete LU factorization of matrix A would be:
L = [1 0 0, 2 1 0, 1 1 1]
U = [1 2 3, 0 -1 -6, 0 0 -3]
It's important to note that if A is a singular matrix or has pivot elements of zero, additional steps like partial pivoting might be necessary. However, in this case, we didn't need partial pivoting since all the pivot elements are non-zero during the process.