Question

Find an LU factorization of the matrix A (with L unit lower triangular). A=

⎣
⎡
​

−3
6
6
​

0
3
15
​

4
−4
21
​

⎦
⎤
​

L=
U=
​

Answer 1

An LU factorization involves decomposing a matrix into a lower triangular matrix L and an upper triangular matrix U. The process uses elementary row operations to achieve this form, and the matrix A is rewritten as the product of L and U.

Step-by-step explanation:

The student has asked for an LU factorization of a given matrix A. LU factorization is a method where a matrix A is decomposed into two matrices - L (a lower triangular matrix with a diagonal of ones) and U (an upper triangular matrix). The process involves using elementary row operations to rewrite matrix A into this form.

We start with the original matrix A:

A =
-3 6 6
0 3 15
4 -4 21

First, we need to transform A into an upper triangular matrix (U). During this process, we perform row operations and keep track of these operations in the form of matrix L.

Then, by applying the appropriate row operations, we can find L and U which satisfy the equation A = LU. It is important to note that this operation may require partial pivoting if there are zeros in the pivot position.