Let Un×n be an upper triangular matrix of rank n. If any arithmetic operation takes 1µ...

Let Un×n be an upper triangular matrix of rank n. If any arithmetic operation takes 1µ second on a computing resource,

compute the time taken to solve the system Ux = b, assuming it has a unique solution. What would be the time taken if Un×n is lower triangular

Solutions

Expert Solution

Given that U is a square matrix with n rows and n columns.

It is a uppee triangular matrix. It means all the elements below the diagonal of the matrix are zeros. Rank of matrix U is n.

For the system Ux=b, as U has n rows, we have n no.of values of x from x1,x2,x3,......,xn.

Given that on computation resource, it take one micro second for one arithmetic operation. Here, in a computation resource, to solve a problem, it may require n number of steps to solve the system.

In this case, time taken is number of steps multiplied by 1micro second.

As Ux=b has n values of x, we will get n number of equations fron Ux=b to find n values of x.

So, time to solve the system = number of equations * time for each srep.

I.e total time = (n)*(1micro sec) = n micro seconds

For a lower triangular matrix, all the elements above the diagonal are zeros. To solve the system Ux=b, we get n equations to solve to find n values of x.

So for lower triangular matrix with the system Ux=b, total time to solve is same as for the upoer triangular matrix.

Hence, time for solving Ux=b, where U is either upper triangular or lower triangular matrix is n microseconds.

Leave a comment for any queries. Upvote if u like it.

Thank you.

Colby Messinger answered 11 months ago

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Question

Let Un×n be an upper triangular matrix of rank n. If any arithmetic operation takes 1µ...

Solutions

Expert Solution

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