Math Challenges

Submissions for Problem #37

Problem #37

Find an LU factorization of the given matrix (L a lower triangular matrix, U an echelon form)

\[ \begin{pmatrix}3 & -1 & 2\\ -3 & -2 & 10\\ 9 & -5 & 6\end{pmatrix} \]

zapwai

Solution: \( L=\begin{pmatrix}1 & 0 & 0\\ -1 & 1 & 0\\ 3 & \frac23 & 1\end{pmatrix}U=\begin{pmatrix}3 & -1 & 2\\ 0 & -3 & 12\\ 0 & 0 & -8\end{pmatrix} \)

Explanation:
These are found by doing row operations (subtracting a multiple of row 1, or row 2 in the last step) to matrix M to make it look like U. At each step we record the negative of the scalar used in the appropriate cell of L. e.g. -3 is -1 times 3, so we record a -1. 9 is 3 times 3, so we record a 3.