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# Matrices : Gaussian Elimination

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How many multiplications are necessary to find the determinants of matrices which are (2,2) (3,3) and (4,4)?

The number of multiplications for an n,n matrix may be found in terms of the number for (n-1) X (n-1) matrix. Find this formula and use it to obtain the number of multiplications for a 10,10 matrix.

For an nXn matrix the number of multiplications is roughly proportional to n! when n is sufficiently large - find the constant of proportionality from your numerical results above, hence find out approximately how many multiplications and divisions (operations) are needed to use Cramer's rule to solve the system Ax=b Where A is nXn.

How long will a computer take to do such a computation for a 15x15 matrix if it uses 1 secound to excute 10^6 operations? Compare your answer with the time taken using Gaussian Elimination for which the operation count is n(n^2+3n-1)/6.

Use Gaussian Elimination to solve the equations,

(7 -1 -1) (x) (3)
(3 2 1) (y) = (4)
(5 2 3) (z) (20)

I got x = 1
Y = -3
Z = 7

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#### Solution Preview

How many multiplications are necessary to find the determinants of matrices which are (2,2) (3,3) and (4,4).
number of multiplications of (2,2) is 2! = 2*1 = 2
number of multiplications of (3,3) is 3! = 3*2*1 = 6
number of multiplications of (4,4) is 4! = 4*3*2*1 = 24

The number of multiplications for an n,n matrix may be found in terms of the number for (n-1) X (n-1) matrix. Find this formula and use it to obtain the number of multiplications for a 10,10 matrix

number of multiplications of (10,10) is 10! = 10*9*...*3*2*1 = ...

#### Solution Summary

Gaussian Elimination and Calculation Time are analyzed. All work is shown.

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