$(5$ marks$)$

Let A $=$

$$

$$

$$

$$

$$

a$11$ a$12$ a$1$n

$0$ a$22$ a$2$n

$............$

$00$ ann

$$

$$

$$

$$

$$ be an n$$n invertible upper triangular matrix.

Find a diagonal matrix D $=$

$$

$$

$$

$$

$$

d$100$

$0$ d$20$

$............$

$00$ dn

$$

$$

$$

$$

$$

and a unit upper triangular matrix U $=$

$$

$$

$$

$$

$$

$1$ u$12$ u$1$n

$01$ u$2$n

$............$

$001$

$$

$$

$$

$$

$$ such that

A $=$ D U $.$

Is the factorization A $=$ D U possible if A is not invertible? Justify your answer