Using Matlab

Enter the matrix A = [2 1 4; 6 2 13; 4 7 4]

(a) Determine elementary matrices of Type III E1, E2, E3 (give names to these elementary matrices so that you can use them again later) such that E3E2E1A = U with U an upper triangular matrix. (Hint: E1 should turn the element in position (2,1) into a 0, E2 should turn the element in position (3,1) into a zero. Once you have found the matrices E1 and E2, compute the product E2E1A. Based on the result, determine the matrix E3 that turns the element in position (3,2) into a zero.)

(b) Compute the product L = E1^1 * E2^-1 * E3^-1 . The matrix L is lower triangular with ones on the diagonal. Verify that A = LU.

NOTE: From part (a) we have E3E2E1A = U and therefore A = (E3E2E1)^-1U = (E1^1 * E2^-1 * E3^-1)U. Since the elementary matrices and their inverses are lower triangular and have 1s along the diagonal, as is always true for elementary matrices of the third type, it follows that L is also lower triangular with 1s along the diagonal