Below is an awkward algorithm to solve the above problem.

ComputeSumLowerTriangular$($L$,$n$)$

inputs: L is an nxn matrix. n is the dimension of the matrix

output: a real number equal to

$1$: sum $=0$

$2$: for i $=$ n to $1$

$3$: psum $=0$

$4$: for j $=1$ to i

$5$: psum $=$ psum $+$ L$[$i$,$j$]$

$6$: sum $=$ sum $+$psum

$7$: return sumLet us call tj the number of comparisons performed by the inner $$for loop$$ in Line $4$ for a given value of j$.$ Fill in this table $($Justify how you find tj using exactly the same steps and sentence pattern shown below for j $=1)$:

i ti

n when i $=$ n$,$ the inner for loop is "for j $=1$ to n$",$ the number of comparisons is $....$ n$.$ The number of comparisons is n due to when the outer loop variable i is set to n$,$ the inner loop variable j is comparing all values from $1$ to n$.$

n$-1$ when i $=$ n$-1,$ the inner for loop is "for j $=...$ to $....",$ the number of comparisons is $....$n$-2.$ The number of comparisons is n$-2$ due to when the outer loop variable i is set to n$-1,$ the inner loop variable j is being compared to all values from $1$ to n$-1$

n$-2($follow the same patter as above$)$ when i $=$ n$-2,$ the inner for loop is $.........,$ the number......

k when i $=$ k$,$ the inner for loop is $....,$ the number $....$

$1$ when i $=1,$ the inner for loop is $....$