Let $A = \begin{pmatrix}-1 & 0 & 3 & 1 \\ -2 & -2 & 9 & 4 \\ 0 & 6 & -7 & -5 \\ -1 & 0 & 3 & 3 \\ \end{pmatrix}$, $E_1 = \begin{pmatrix}1 & 0 & 0 & 0 \\ -2 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$ and $E_2 = \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 3 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$. Then answer the following questions.

  1. Find $E_1^{-1}$.

    $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ -\frac{1}{2} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ -2 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ -k & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$

  2. Find $A_2 = E_2 E_1 A$.

    $\displaystyle \begin{pmatrix}-1 & 0 & 3 & 1 \\ 0 & 2 & 3 & 2 \\ 0 & 0 & 2 & 1 \\ -1 & 0 & 3 & 3 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}-1 & 0 & 3 & 1 \\ 0 & -2 & 3 & 2 \\ 0 & 0 & 2 & 1 \\ -1 & 0 & 3 & 3 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}-1 & 0 & 3 & 1 \\ 0 & -2 & 3 & 2 \\ 0 & 0 & -1 & 2 \\ -1 & 0 & 3 & 3 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}-1 & 0 & 3 & 1 \\ 0 & -2 & 3 & 2 \\ 0 & 0 & 2 & -1 \\ -1 & 0 & 3 & 3 \\ \end{pmatrix}$

  3. Find the elementary matrix $E_3$ so that $U = E_3 A_2$ becomes an upper triangular matrix.

    $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - 1 & 0 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 4 & 0 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ \end{pmatrix}$

  4. Find the upper triangular matrix $U$ in the LU decomposition of $A$.

    $\displaystyle \begin{pmatrix}-1 & 0 & 3 & 1 \\ 0 & -2 & 3 & 2 \\ 0 & 0 & 2 & -1 \\ 0 & 0 & 0 & 2 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}-1 & 0 & 3 & 1 \\ 0 & 2 & 3 & 2 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}-1 & 0 & 3 & 1 \\ 0 & -2 & 3 & 2 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}-1 & 0 & 3 & 1 \\ 0 & -2 & 3 & 2 \\ 0 & 0 & -1 & 2 \\ 0 & 0 & 0 & 2 \\ \end{pmatrix}$

  5. Find the lower triangular matrix $L$ in the LU decomposition of $A$.

    $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ -k & 1 & 0 & 0 \\ 0 & 3 & 1 & 0 \\ -1 & 0 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ -2 & 1 & 0 & 0 \\ 0 & 3 & 1 & 0 \\ -1 & 0 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 0 & -3 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 & 0 \\ -k & 1 & 0 & 0 \\ 0 & -3 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ \end{pmatrix}$


Department of Mathematics
Last modified: 2025-07-19