Exercise 1 (4 points) In this exercise you will create the three types of elementary matrices, analyze their properties, and perform elementary row-operations on matrices by using the elementary matrices (a) **Write three functions in MATLAB function E1-elel (n,r.i,j) fiunction E2-ele2 (n,i, j) function E3-ele3 (n, j,k) The functions create elementary nxn matrices that correspond to the elementary row operations, namely, row replacement, row interchanging, and row scaling You should start with the identity matrix eye(n) Matrix El is obtained from eye(n) by replacing (row j) with [(rowj) + (rowi).r] Matrix E2 is obtained from the matrix eye(n) by interchanging rows i andj Matrix E3 is obtained from eye(n) by multiplying rowj by k. *Type the functions in your diary file type elel e ele2 type ele3 Part I **Run each of the three created function for n-4; r-3; i-2;j-4; k-5; Your outputs must be three matrices E1, E2, E3 for the indicated variables and you will be working with them in this part of the Exercise %Write a comment how each of these matrices was generated from the matrix eye(4) **Calculate the determinant of each of the elementary matrices E1, E2, E3 det(E1), det(E2), and det(E3). Output them % Comment within your diary file whether the determinants, which you have calculated. are different form the determinant of the original identity matrix and what is the difference **Type the matrix M-111 1: 2 2 2 2: 3 3 3 3: 4 4 4 4] in your diary file ** Left multiply matrix M by each of the output matrices E1, E2, E3 (one at a time). Output he three products % Comment on how matrix M has been modified after performing each multiplication. Part II **Within your diary file, create a matrix A by starting with eye(5) and performing the following sequence of elementary row-operations: first, replacing (row 2) with [(row 2)+ (row 4)-3], then, interchanging rows 1 and 3, and, finally, scaling row 5 by 2. To complete this part, you, should, first, create the required elementary matrices for the corresponding sets of variables (the variables are specific for each function!) and, then, calculate matrix A by using left multiplication of eye(5) by the created elementary matrices in the indicated order % Explain in your diary file a reason why the obtained matrix A is invertible **Find matrix C, which is the inverse of A, by using command C-inv(A). Output the result. *Then, recalculate C by multiplying (in the correct order!) the inverses of the elementary matrices whose product is matrix A. (Please watch for the order of multiplication!) Output the result Note: You should receive the same matrix C-Aby using two different methods
