Problems

01

Row reduction of a product

Suppose $A=BC$ and $B$ is invertible. Prove that any sequence of row reduction operations which reduces $B$ to $I_n$ will also reduce $A$ to $C$. Can you relate this fact to the LU decomposition?

02

Inverse of $A$ using LU

Compute the inverse of $A$ from Problem 09-02 using the LU factorization computed in that problem, in other words by computing $U^{-1}{L}^{-1}$. Now compute $A^{-1}$ using the method at the end of the section on preimages of matrices: perform row reduction on the augmented matrix $\left(\begin{array}{rr}A& I_4\end{array}\right)$. Which method do you like better?

03

Row reduction with pivoting

Perform row reduction with pivoting upon the matrix $A=\left(\begin{array}{rccc}2& 0& 0& 1\\ 4& 0& 0& 0\\ 4& 0& 3& 9\\ -6& 0& 24& 49\end{array}\right)$ in order to find all solutions $\mathbf{x}$ to the equation $A\mathbf{x}=\mathbf{b}$ with $\mathbf{b}=(1,1,1,1)$. (You can use the augmented matrix; no need to compute the full $PA=LU$ decomposition.)