TITLE: Upper and Lower Triangular Matrices
QUESTION [8 upvotes]: Given the matrix A=$ \left( \begin{array}{ccc}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8\\
1 & -1 & 2 & 3 \\
2 & 1 & 1 &2\end{array} \right) $, write it in the $L_{4 \times 4}U_{4 \times 4}$, where L is the lower triangular matrix and U is the upper triangular matrix.
To be honest, I don't even understand what the question is asking of me, however I do know what upper and lower triangular matrices are.
Thanks in advance for your help.

REPLY [15 votes]: Let's go step by step.  We want an equation of the following form:
$$\begin{pmatrix}1&2&3&4\\5&6&7&8\\1&-1&2&3\\2&1&1&2\end{pmatrix}=\begin{pmatrix}\star&0&0&0\\\star&\star&0&0\\\star&\star&\star&0\\\star&\star&\star&\star\end{pmatrix}\begin{pmatrix}\star&\star&\star&\star\\0&\star&\star&\star\\0&0&\star&\star\\0&0&0&\star\end{pmatrix}$$
From the first column and first row of our known matrix, it's not too hard to see that we can start with this:
$$\begin{pmatrix}1&2&3&4\\5&6&7&8\\1&-1&2&3\\2&1&1&2\end{pmatrix}=\begin{pmatrix}1&0&0&0\\5&\star&0&0\\1&\star&\star&0\\2&\star&\star&\star\end{pmatrix}\begin{pmatrix}1&2&3&4\\0&\star&\star&\star\\0&0&\star&\star\\0&0&0&\star\end{pmatrix}$$
Next, we can choose the diagonal elements of our upper triangular matrix to be $1$, and fill in the lower triangular matrix column by column:
$$\begin{pmatrix}1&2&3&4\\5&6&7&8\\1&-1&2&3\\2&1&1&2\end{pmatrix}=\begin{pmatrix}1&0&0&0\\5&-4&0&0\\1&-3&\star&0\\2&-3&\star&\star\end{pmatrix}\begin{pmatrix}1&2&3&4\\0&1&\star&\star\\0&0&1&\star\\0&0&0&1\end{pmatrix}$$
$$\begin{pmatrix}1&2&3&4\\5&6&7&8\\1&-1&2&3\\2&1&1&2\end{pmatrix}=\begin{pmatrix}1&0&0&0\\5&-4&0&0\\1&-3&5&0\\2&-3&1&\star\end{pmatrix}\begin{pmatrix}1&2&3&4\\0&1&2&\star\\0&0&1&\star\\0&0&0&1\end{pmatrix}$$
$$\begin{pmatrix}1&2&3&4\\5&6&7&8\\1&-1&2&3\\2&1&1&2\end{pmatrix}=\begin{pmatrix}1&0&0&0\\5&-4&0&0\\1&-3&5&0\\2&-3&1&7/5\end{pmatrix}\begin{pmatrix}1&2&3&4\\0&1&2&3\\0&0&1&8/5\\0&0&0&1\end{pmatrix}$$