Problems

01

LU factorization and solving $A\mathbf{x}=\mathbf{b}$

Compute the LU factorization of $A$:

$$A=\left(\begin{array}{rccc}1& -2& -4& -3\\ 2& -7& -7& -6\\ -1& 2& 6& 4\\ -4& -1& 9& 8\end{array}\right).$$

Use this to find the unique value of $\mathbf{x}$ that satisfies $A\mathbf{x}=\mathbf{b}$, where $\mathbf{b}=(1,7,0,3)$. (You should perform the elimination and back-substitution operations on the rows of $\mathbf{b}$ that are dictated by the LU decomposition of $A$.) Also: what is the determinant of $A$?