KEVOS® MAKEOS · Mathematics
3 × 3 Matrix Determinant, Adjoint and Inverse
What are the determinant and inverse of this 3 × 3 matrix?
Determinant, adjoint (adjugate) and inverse of a 3 × 3 matrix by cofactors, as set out in the handbook’s matrix operations, with a check that the matrix times its inverse returns the identity.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| a₁₁ | Row 1, column 1. | |
| a₁₂ | Row 1, column 2. | |
| a₁₃ | Row 1, column 3. | |
| a₂₁ | Row 2, column 1. | |
| a₂₂ | Row 2, column 2. | |
| a₂₃ | Row 2, column 3. | |
| a₃₁ | Row 3, column 1. | |
| a₃₂ | Row 3, column 2. | |
| a₃₃ | Row 3, column 3. |
Method and formulas
- Determinant by expansion along the first row.
- Adjoint = transpose of the matrix of cofactors.
- Inverse = adjoint ÷ determinant (exists only when the determinant is not zero).
- Example: the default matrix has a determinant of 1.
Limitations
- A determinant near zero relative to the entries means an ill-conditioned matrix; small input errors give large changes in the inverse.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Invert a transformation or stiffness matrix
- Check whether a set of equations has a unique solution
- Teach or verify hand matrix work
Who it is for
- Engineers
- Technicians
- Engineering students
What you get
- Determinant
- Adjoint
- Inverse
Interpreting the result
- Multiply the inverse by the right-hand side vector to solve the three equations.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 122–125. Topic reference; no source tables or recommended settings are reproduced.
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