Matrix Calculator: Determinant, Inverse, Transpose and Eigenvalues

Determinant, trace, transpose, inverse and eigenvalues of a 2x2 or 3x3 matrix, with the working shown.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose 2x2 or 3x3 and type in the numbers. The determinant is worked out by expanding along the first row. If it is not zero, the matrix has an inverse, found from the cofactors (the adjugate divided by the determinant). A zero determinant is flagged as singular.

The eigenvalues come from the characteristic equation. For a 3x3 matrix it is a cubic, solved exactly, and any complex eigenvalues are shown as a plus or minus b i. The sum of the eigenvalues must equal the trace and their product the determinant, which is a quick check.

Matrices appear in structural stiffness, stress transformation, vibration modes and control. To solve a set of equations directly, use the simultaneous equations calculator.

Formulas used

  • \det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc
  • A^{-1} = \frac{1}{\det A}\operatorname{adj}(A)
  • \det(A - \lambda I) = 0

Worked example

With the default numbers in the calculator above:

  1. The matrix: 3 x 3 matrix
  2. Determinant (expanding along the first row): 18.00
  3. Trace (sum of the diagonal): 9.00
  4. Transpose: [2, 1, 0] [1, 3, 1] [0, 1, 4]
  5. Cofactor matrix: [11, -4, 1] [-4, 8, -2] [1, -2, 5]
  6. Inverse (adjugate divided by the determinant): [0.61111, -0.22222, 0.055556] [-0.22222, 0.44444, -0.11111] [0.055556, -0.11111, 0.27778]
  7. Characteristic equation: lambda^3 - 9.00 lambda^2 + 24.00 lambda - 18.00 = 0
  8. Eigenvalues: 1.27, 3.00, 4.73

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Row 1, column 1 2; Row 1, column 2 1; Row 1, column 3 1; Row 2, column 1 0; Row 2, column 2 3; Row 2, column 3 1; Row 3, column 1 0; Row 3, column 2 0; Row 3, column 3 4.

  1. The matrix: 3 x 3 matrix
  2. Determinant (expanding along the first row): 24.00
  3. Trace (sum of the diagonal): 9.00
  4. Transpose: [2, 0, 0] [1, 3, 0] [1, 1, 4]
  5. Cofactor matrix: [12, 0, 0] [-4, 8, 0] [-2, -2, 6]
  6. Inverse (adjugate divided by the determinant): [0.5, -0.16667, -0.083333] [0, 0.33333, -0.083333] [0, 0, 0.25]
  7. Characteristic equation: lambda^3 - 9.00 lambda^2 + 26.00 lambda - 24.00 = 0
  8. Eigenvalues: 2.00, 3.00, 4.00

Common questions

What does a zero determinant mean?

The matrix is singular: it has no inverse, and the equations it represents are dependent or have no single solution.

What are eigenvalues used for?

They give the principal stresses of a stress tensor, the natural frequencies of a vibrating system, and the stability of a control system.

How do I solve A x = b?

Use the simultaneous equations calculator, or multiply b by the inverse found here.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Kreyszig, Advanced Engineering Mathematics (linear algebra); Strang, Introduction to Linear Algebra.

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