Finding the $UL$ decomposition of a matrix. (Linear Algebra) Find the $A=UL$ decomposition of the following matrix.(Note the letters $UL$)

$$
A =
\left[ {\begin{array}{cc}
a &amp; b+c &amp; b \\
b &amp; b+c&amp; b \\
c &amp; c &amp; b
\end{array} } \right]
$$

To find $L$ i took the first row subtract the second row to get $
 =
\left[ {\begin{array}{cc}
a-b &amp; 0 &amp; 0 \\
b &amp; b+c&amp; b \\
c &amp; c &amp; b
\end{array} } \right]
$

Next, I took the second row and subtract the third row from it to get $
 =
\left[ {\begin{array}{cc}
a-b &amp; 0 &amp; 0 \\
b-c &amp; b&amp; 0 \\
c &amp; c &amp; b
\end{array} } \right]
$.   Hence  $$
 L=
\left[ {\begin{array}{cc}
a-b &amp; 0 &amp; 0 \\
b-c &amp; b&amp; 0 \\
c &amp; c &amp; b
\end{array} } \right]
$$ Form $L$ i found my $U$ to be 
$$
U =
\left[ {\begin{array}{cc}
1 &amp; 1 &amp; 1 \\
0 &amp; 1&amp; 1 \\
0 &amp; 0 &amp; 1
\end{array} } \right]
$$

Is my working correct. Could anyone explain. Thanks
I see what you've done but I think the top right entry of your upper triangular matrix should be zero, since you're not really subtracting the third row from the first right?
I see what you've done but I think the top right entry of your upper triangular matrix should be zero, since you're not really subtracting the third row from the first right?