SOLUTION: Consider the following invertible matrix A, and vector B. A  =  2 4 4 3 8 7 10 10 5 2 1 6 8 7 5 5 5 4 10 5 5 1 7 6 3 , B = 3 5 7 10 6 . Use Cra

Algebra ->  Matrices-and-determiminant -> SOLUTION: Consider the following invertible matrix A, and vector B. A  =  2 4 4 3 8 7 10 10 5 2 1 6 8 7 5 5 5 4 10 5 5 1 7 6 3 , B = 3 5 7 10 6 . Use Cra      Log On


   

Question 1084344: Consider the following invertible matrix A, and vector B.
A  = 
2 4 4 3 8
7 10 10 5 2
1 6 8 7 5
5 5 4 10 5
5 1 7 6 3
, B =
3
5
7
10
6
.
Use Cramer's Rule to solve the following.
(a) Solve the equation AX = B for the variable x3 .
(b) Solve the equation AX = B for the variable x4 .

Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.
About Cramer's rule read the lessons
    - Determinant of a 3x3 matrix
    - Co-factoring the determinant of a 3x3 matrix
    - HOW TO solve system of linear equations in three unknowns using determinant (Cramer's rule)
    - Solving systems of linear equations in three unknowns using determinant (Cramer's rule)
    - Solving word problems by reducing to systems of linear equations in three unknowns
    - The tricks to solve some word problems with three and more unknowns using mental math
    - Solving systems of non-linear equations in three unknowns using Cramer's rule
in this site.


Now let me reveal you one secret.

For matrices of dimensions 4, 5 and higher, practically nobody in nova days applies Cramer's rule with handle/manual calculations.
People use calculators or computer software. Very popular are internet sites that offer online solutions (solvers) for free . . . (!)

One such solver you can find at the link
http://matrix.reshish.com/cramSolution.php

http://matrix.reshish.com/cramSolution.php


I used it to get the solution presented below: 

Your matrix

№	X1	X2	X3	X4	X5	b
1	2	4	4	3	8	3
2	7	10	10	5	2	5
3	1	6	8	7	5	7
4	5	5	4	10	5	10
5	5	1	7	6	3	6

Write down the main matrix and find its determinant

№	X1	X2	X3	X4	X5
1	2	4	4	3	8
2	7	10	10	5	2
3	1	6	8	7	5
4	5	5	4	10	5
5	5	1	7	6	3
D = -16605

Replace the 1st column of the main matrix with the solution vector and find its determinant

№	X1	X2	X3	X4	X5
1	3	4	4	3	8
2	5	10	10	5	2
3	7	6	8	7	5
4	10	5	4	10	5
5	6	1	7	6	3
D1 = 0

Replace the 2nd column of the main matrix with the solution vector and find its determinant

№	X1	X2	X3	X4	X5
1	2	3	4	3	8
2	7	5	10	5	2
3	1	7	8	7	5
4	5	10	4	10	5
5	5	6	7	6	3
D2 = 0

Replace the 3rd column of the main matrix with the solution vector and find its determinant

№	X1	X2	X3	X4	X5
1	2	4	3	3	8
2	7	10	5	5	2
3	1	6	7	7	5
4	5	5	10	10	5
5	5	1	6	6	3
D3 = 0

Replace the 4th column of the main matrix with the solution vector and find its determinant

№	X1	X2	X3	X4	X5
1	2	4	4	3	8
2	7	10	10	5	2
3	1	6	8	7	5
4	5	5	4	10	5
5	5	1	7	6	3
D4 = -16605

Replace the 5th column of the main matrix with the solution vector and find its determinant

№	X1	X2	X3	X4	X5
1	2	4	4	3	3
2	7	10	10	5	5
3	1	6	8	7	7
4	5	5	4	10	10
5	5	1	7	6	6
D5 = 0

x1 = D1 / D = 0 / (-16605) = 0
x2 = D2 / D = 0 / (-16605) = 0
x3 = D3 / D = 0 / (-16605) = 0
x4 = D4 / D = (-16605) / (-16605) = 1
x5 = D5 / D = 0 / (-16605) = 0

Solution set:

x1 = 0
x2 = 0
x3 = 0
x4 = 1
x5 = 0