BIZOMAT 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. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: a₁₁
- What it is
- The element in row 1, column 1 of the matrix A: the top-left entry.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Row first, then column: aᵢⱼ sits in row i, column j. Swapping them enters the transpose, which has the same determinant but a transposed inverse.
What this means: a₁₂
- What it is
- The element in row 1, column 2 of the matrix A: the middle entry of the top row.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: a₁₃
- What it is
- The element in row 1, column 3 of the matrix A: the top-right entry.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: a₂₁
- What it is
- The element in row 2, column 1 of the matrix A: the left entry of the middle row.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: a₂₂
- What it is
- The element in row 2, column 2 of the matrix A: the centre entry.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: a₂₃
- What it is
- The element in row 2, column 3 of the matrix A: the right entry of the middle row.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: a₃₁
- What it is
- The element in row 3, column 1 of the matrix A: the bottom-left entry.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: a₃₂
- What it is
- The element in row 3, column 2 of the matrix A: the middle entry of the bottom row.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: a₃₃
- What it is
- The element in row 3, column 3 of the matrix A: the bottom-right entry.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
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.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Useful alongside · BIZOMAT original book
Strain and Hooke's law in two and three dimensions
In Stress Transformation by the Numbers
What it covers: What normal and shear strains are, how they transform like stresses, how Hooke's law links stress and strain in three directions, and why a single gauge reading times E is usually the wrong stress.
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Open the 3 × 3 Matrix Determinant, Adjoint and Inverse calculator