\item
Prove that the $LU$ decomposition, if exists, is unique.

\item
What is Cholesky decomposition? 
When does a matrix admit Cholesky decomposition?
(Need not prove your statement).

\item
Write a pseudo-code for Cholesky decomposition for
a matrix that satisfies the requirement from problem 2
and is also a tridiagonal matrix.
Give the number
of multiplications/divisions to leading order
(assuming that a square root amounts to 10 multiplications).
Give your answer as $CN^p$, find $C$ and $p$.

\item
Let $A$ be an $20 \times 20$ tridiagonal matrix with 
$a_{ii}=5$, $a_{i+1,i}=a_{i,i+1}=2$ and $a_{ij}=0$ otherwise.
Show that Gaussian elimination on $A$ is the same as 
Gaussian elimination with partial pivoting.

\item
Let $b = (1,\cdots,1)'$ and $A$ as above.
Use the octave built-in function to get $LU$ decomposition for $A$.
Find $z =L^{-1} b$ via forward/backward 
(you have to decide which) substitution.
Then compute max(abs($L*z-b$)). 
Copy your code (carefully) and max(abs($L*z-b$)) on the answer sheet.