LINEAR ALGEBRA PROJECT USING MATLAB) Part 1 2)Investigate products of lower tria

Question

LINEAR ALGEBRA PROJECT USING MATLAB)

Part 1

2)Investigate products of lower triangular matrices which have all the diagonal entries equal to 1. Such a matrix is called a unit lower triangular matrix.For example you could type

n=2

L1=tril(rand(n),-1)+eye(n),L2=tril(rand(n),-1)+eye(n),L1*L2,L2*L1

Execute this line several times and inspect the result each time .Repeat with n=3,4,5

a) For each pair of L1 and L2 , what entries of LIL2 and L2L1 appear to be the same?

b) Prove that the product of any two n*n unit lower triangular matrices will be a unit lower triangular matrix ( you need to prove here that each diagonal entry of the product is 1, why?

c

Explanation / Answer

Execute the second line several times and inspect the result each time. Repeat with n = 3,4,5. The easy way is to press the up arrow on your keyboard to retrieve the second line, and then press [Enter] to execute it again. Each execution of rand(n)produces a new n n × matrix with random number entries. The function tril puts zeros in the upper triangle, and tril(rand(n)) produces a random lower triangular matrix. (a) (hand) For each pair L1 and L2 , what entries of L L1 2 and L L

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