Find the $A=UL$ decomposition of the following matrix.(Note the letters $UL$)
$$ A = \left[ {\begin{array}{cc} a & b+c & b \\ b & b+c& b \\ c & c & b \end{array} } \right] $$
To find $L$ i took the first row subtract the second row to get $ = \left[ {\begin{array}{cc} a-b & 0 & 0 \\ b & b+c& b \\ c & c & 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 & 0 & 0 \\ b-c & b& 0 \\ c & c & b \end{array} } \right] $. Hence $$ L= \left[ {\begin{array}{cc} a-b & 0 & 0 \\ b-c & b& 0 \\ c & c & b \end{array} } \right] $$ Form $L$ i found my $U$ to be $$ U = \left[ {\begin{array}{cc} 1 & 1 & 1 \\ 0 & 1& 1 \\ 0 & 0 & 1 \end{array} } \right] $$
Is my working correct. Could anyone explain. Thanks