The algorithm shown below factors an $nn$ matrix A into $LU$ by Dolittle decomposition and

returns the result. The algorithm does this by storing $L$ and $U$ in the space originally occupied by $A$

thereby economizing storage space.

a$)$ Represent the algorithm as a flow chart.

b$)$ Write a Matlab program which uses this algorithm to factor the matrix $A=\left[\begin{array}{cc}1& 2\\ 3& 9\end{array}\right].$ In writing

your program, be careful not to store more than one $nn$ matrix at any given time which

would defeat the purpose of the economization.

c$)$ Factor the matrix in part $($b$)$ manually following this algorithm. Show the expression and the

numerical value of each variable as it is computed.

d$)$ Determine the matrices $L$ and $U$ from the result obtained in part $($c$).$

a$)$ Find the inverse and determinant of the matrix in problem $(1$b$)$ using Gauss$-$Jordan elimination.

b$)$ Find the condition number of the matrix based on the Euclidean norm.

c$)$ How many digits of precision would you expect to lose due to ill$-$conditioning when solving the

system of equations $Ax=b?$