Modify the LU Factorization Algorithm so that it can be used to solve a linear system, and then solve the following linear systems.
a. x1 − x2 = 2,
2x1 + 2x2 + 3x3 = −1,
−x1 + 3x2 + 2x3 = 4
b. 1/3 x1 + 1/2 x2 - 1/4 x3 = 1,
1/5 x1 + 2/3 x2 + 3/8 x3 = 2,
2/5 x1 - 2/3 x2 + 5/8 x3 = −3.
b. 2x1 + x2 = 0,
−x1 + 3x2 + 3x3 = 5,
2x1 − 2x2 + x3 + 4x4 = −2,
−2x1 + 2x2 + 2x3 + 5x4 = 6
d. 2.121x1 − 3.460x2 + 5.217x4 = 1.909,
5.193x2 − 2.197x3 + 4.206x4 = 0,
5.132x1 + 1.414x2 + 3.141x3 = −2.101,
−3.111x1 − 1.732x2 + 2.718x3 + 5.212x4 = 6.824