1
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ It can be factored using $A = PLU$, where $P$ is a permutation matrix. $\endgroup$ – Amzoti Nov 19 '13 at 22:50
3
$\begingroup$

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

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 $\det M \not= 0$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.