KEVOS® MAKEOS · Mathematics
Simultaneous Equations and 3 × 3 Matrix
What solves these two or three simultaneous linear equations, and what are the determinant and inverse?
Solution of two or three simultaneous linear equations, with the determinant, inverse and transpose of the coefficient matrix.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Size | choice | Two or three equations (the third row and column are ignored for two). |
| Equation 1 | ||
| Coefficient of x1 | Coefficient of x1 in equation 1. | |
| Coefficient of x2 | Coefficient of x2 in equation 1. | |
| Coefficient of x3 | Coefficient of x3 in equation 1. | |
| Right-hand side | Constant on the right-hand side of equation 1. | |
| Equation 2 | ||
| Coefficient of x1 | Coefficient of x1 in equation 2. | |
| Coefficient of x2 | Coefficient of x2 in equation 2. | |
| Coefficient of x3 | Coefficient of x3 in equation 2. | |
| Right-hand side | Constant on the right-hand side of equation 2. | |
| Equation 3 | ||
| Coefficient of x1 | Coefficient of x1 in equation 3. | |
| Coefficient of x2 | Coefficient of x2 in equation 3. | |
| Coefficient of x3 | Coefficient of x3 in equation 3. | |
| Right-hand side | Constant on the right-hand side of equation 3. | |
Method and formulas
- Determinant by cofactor expansion; the system has a unique solution when it is not zero.
- Inverse = adjugate ÷ determinant; solution x = A⁻¹ b (Cramer’s rule).
- Transpose swaps rows and columns.
- Example: the default system gives x₁ = 2, x₂ = 3, x₃ = -1.
Limitations
- Nearly singular matrices (determinant close to zero) give unreliable solutions.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Solve force balance and reaction equations
- Invert small matrices for transformations
- Check hand matrix calculations
Who it is for
- Engineers
- Technicians
- Engineering students
What you get
- Solution x₁, x₂ (, x₃)
- Determinant
- Inverse and transpose of the matrix
Interpreting the result
- Substitute the solution back into the original equations as a check.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 122–127. Topic reference; no source tables or recommended settings are reproduced.
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