Solve by hand without using your calculator. Show all intermediate steps.

Let

L $=$

$$

$$

$$

$$

$1000$

$1200$

$1/23/220$

$1411$

$$

$$

$$

$$

$($i$)(6$ pts$.)$ Solve Lx $=$ b for an arbitrary vector b$.$ Display L$1.$

$($ii$)(4$ pts$.)$ Express the matrix

G $=$

$$

$$

$$

$$

$2800$

$21631$

$1660$

$2000$

$$

$$

$$

$$

in the form G $=$ PLD$,$ where P is a permutation matrix, D is a diagonal matrix and L is as given

above. Determine G$1.$

$($iii$)(6$ pts$.)$ If

A $=$

$$

$$

$$

$$

$2111$

$2517$

$1522$

$211513$

$$

$$

$$

$$

and b $=[12816$c $]$T $,$ solve the equation Ax $=$ b using Gaussian elimination $($without row

interchanges$).$ Here c is an arbitrary real constant.

$($iv$)(4$ pts$.)$ For A and L as above, show that A $=$ LU$,$ where U is an upper$-$triangular matrix.

$($Hint: You know L$1$ from part $($i$).)$