# Showing a matrix can't be factored into unit lower triangular matrix and upper triangular matrix

I'm trying to show the following matrix cannot be factored into the product of a unit lower triangular matrix and an upper triangular matrix.

$$\pmatrix{ 2 & 2 & 1 \\ 1 & 1 & 1 \\ 3 & 2 & 1}$$

I'm trying different row operations, attempting to get A into an upper triangular matrix and can't. Is there explicit way to show that it can't be factored into those two matrices?

• It can be factored using $A = PLU$, where $P$ is a permutation matrix. – Amzoti Nov 19 '13 at 22:50

$$M=\pmatrix{ 2 & 2 & 1 \\ 1 & 1 & 1 \\ 3 & 2 & 1}$$

Suppose it can be factored

$$M=\begin{pmatrix}1&&\\a&1&\\b&c&1\end{pmatrix}\begin{pmatrix}d&e&f\\&g&h\\&&i\end{pmatrix}=\begin{pmatrix}x_{i,j}\end{pmatrix}$$

$2=x_{1,1}=d$ so $d = 2$

$1=x_{2,1}=ad$ so $a=\frac{1}{2}$

$2=x_{1,2}=e$ so $e=2$

$1=x_{2,2}=ea+g=2\frac{1}{2}+g=1+g$ so $g=0$

Which is absurd because $\det M \not= 0$