$($A$)5$ arrays are given with their dimensions: A $30$x$45,$ B $45$x$10,$ C $10$x$30,$ D $30$x$20,$ E $20$x$30.$ What is the optimal order $($parenthesization$)$ of the multiplications $($arrays$)$ to calculate the product of the final multiplication A$*$B$*$C$*$D$*$E $($ie with the fewest operations$)?$ What are the dimensions of the product $($table$)?$

$($B$)$ Describe $($in pseudocode, and with all necessary detail$)$ the algorithm you used to answer $($A$).$ what is its complexity?

$($C$)$ What kind of algorithm is it$?$ Explain why.