.
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