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Your matrix
X1 X2 X3 b
1 2 3 -4 -4
2 1 -2 2 3
3 1 3 -2 1
Write down the main matrix and find its determinant
X1 X2 X3
1 2 3 -4
2 1 -2 2
3 1 3 -2
Δ = -12
Replace the 1st column of the main matrix with the solution vector and find its determinant
X1 X2 X3
1 -4 3 -4
2 3 -2 2
3 1 3 -2
Δ1 = -12
Replace the 2nd column of the main matrix with the solution vector and find its determinant
X1 X2 X3
1 2 -4 -4
2 1 3 2
3 1 1 -2
Δ2 = -24
Replace the 3rd column of the main matrix with the solution vector and find its determinant
X1 X2 X3
1 2 3 -4
2 1 -2 3
3 1 3 1
Δ3 = -36
x1 = Δ1 / Δ = (-12) / (-12) = 1
x2 = Δ2 / Δ = (-24) / (-12) = 2
x3 = Δ3 / Δ = (-36) / (-12) = 3
Solution set:
x1 = 1
x2 = 2
x3 = 3
Solved.
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On Cramer's rule for solving systems of 3 equations in 3 unknowns see 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
in this site.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic
"3x3-Matrices, determinants, Cramer's rule for systems in three unknowns"
In addition, there are many free of charge SOLVERS on a Cramer's rule in the internet.
One of such popular solvers is in this site under the link
https://www.algebra.com/algebra/homework/Matrices-and-determiminant/cramers-rule-3x3.solver
https://www.algebra.com/algebra/homework/Matrices-and-determiminant/cramers-rule-3x3.solver