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Problem: Crout's method 7y+2x=11,3x-y=5 [ Calculator, Method and examples ]

Solution:
Your problem -> Crout's method 7y+2x=11,3x-y=5

Total Equations are 2

2x+7y=11 -> (1)

3x-y=5 -> (2)

Now converting given equations into matrix form
[[2,7],[3,-1]] [[x],[y]]=[[11],[5]]

Now, A = [[2,7],[3,-1]], X = [[x],[y]] and B = [[11],[5]]

LU decomposition : If we have a matrix A, then an upper triangular matrix U can be obtained without pivoting under Gaussian Elimination method, and there exists lower triangular matrix L such that A=LU.

Here A =
 2 7 3 -1

Using Gaussian Elimination method
R_2 larr R_2-(3/2)xx R_1 [:.L_(2,1)=color{blue}{3/2}]

=
 2 7 0 -23/2

:.U =
 2 7 0 -23/2

L is just made up of the multipliers we used in Gaussian elimination with 1s on the diagonal.

:.L =
 1 0 color{blue}{3/2} 1

:. LU decomposition for A is

A =
 2 7 3 -1
=
 1 0 3/2 1
xx
 2 7 0 -23/2
= LU

Now, Ax=B, and A=LU => LUx=B

let Ux=y, then Ly=B =>

 1 0 3/2 1
xx
 y_1 y_2
=
 11 5

  y_1 = 11   3/2y_1 + y_2 = 5 

Now use forward substitution method
From (1)

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