Question

5. Suppose L is a unit-lower-triangular matrix. (a) What can you say about the reduced row echelon form R of L? Be as specific as possible. (b) If you perform the elementary row operations on I that are used to transform L to R, what matrix do you get? Write your answer in terms of I, R, and/or L and basic matrix operations (addition, multiplication, transpose, inverse, etc.). (c) If you perform the same elementary row operations to a matrix A = LU, what matrix do you get? Write your answer in terms of I, R, L, U, and/or A and basic matrix operations (addition, multiplication, transpose, inverse, etc.).

Answer 1

Te Page 5. Suppose L is an is an nxn. unit-lower- treagoelas matrexa 3) LP aq azz Я ang anna! (a) dan make Since we have already n pivots, we in th entries equal to o for é >j by doing elementary row transformationale, we can eliminate each column to get to The fou reduced you echolen for R of L is where I is the inin identity matrin. => RT I

рате Page to 'L (b) Each resor elementary row decansformation is multiplication by an invertible matrin, say Ri Let we deze perfeómed, m- elementary @ row transformation Then IR Rm to such that Rom Rm-i R, L : I where ht invertible L. of 7 Rii Rm are 1 row Take S = Rm Rm-i Ro R, Then s. is invertible because each Ri is invertible for lo 1 sism. S= Rm. Ry is the Son elementary, operations pem performed to to get R. We want to SI =s. We have SL RR = 1 SD SLL I <! El 15 =ť Bij cewe perform the_elementary You Ooperations that are used nd n

to transform 4 to R, we get I. ke, the inverse (c) We have Rm - R, = S = [ g L Rm R, A = Rm R, LU ASLUZ ELU (0 => y dow we perform the same elementary to a Matrin A=20, get operations we U,