2x2 Matrix Transformation Calculator: Determinant, Eigenvalues, Inverse, Rotation, Shear and Projection

How this calculator works

Enter the four numbers of the matrix and a point to move.

The calculator finds the determinant (the area scale), the trace, the eigenvalues and unit eigenvectors, the inverse, and where the matrix sends your point.

It tests whether the matrix is a rotation (and gives the angle), a reflection, a shear, a projection or a scaling.

Formulas used

• \det A = ad - bc
• \lambda = \frac{\mathrm{tr}A\pm\sqrt{\mathrm{tr}^2A - 4\det A}}{2}
• A^{-1} = \frac{1}{\det A}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}
• R^TR = I,\ \det R = 1

Worked example

With the default numbers in the calculator above:

1. Determinant: 0.33
2. What the determinant does to area: Areas are multiplied by 0.33, and the orientation is kept
3. Trace: 1.72
4. Eigenvalues: the factors by which the matrix stretches along its special directions: 1.50 and 0.22
5. Eigenvectors (unit length): the directions that are only stretched: (1.0000, 0.0000) for 1.50 and (0.3317, -0.9434) for 0.22
6. Inverse matrix (it undoes the transformation): [0.67, -1.36; 0.00, 4.55]
7. Where the matrix sends the point you gave: (1.95, 0.22)
8. What kind of transformation it is: A general linear transformation (a mix of stretching, turning and shearing)

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 0.99; Row 1, column 2 0.26; Row 2, column 1 0.26; Row 2, column 2 0.69.

1. Determinant: 0.62
2. What the determinant does to area: Areas are multiplied by 0.62, and the orientation is kept
3. Trace: 1.68
4. Eigenvalues: the factors by which the matrix stretches along its special directions: 1.14 and 0.54
5. Eigenvectors (unit length): the directions that are only stretched: (0.8659, 0.5001) for 1.14 and (0.5001, -0.8659) for 0.54
6. Inverse matrix (it undoes the transformation): [1.12, -0.42; -0.42, 1.61]
7. Where the matrix sends the point you gave: (1.25, 0.95)
8. What kind of transformation it is: A general linear transformation (a mix of stretching, turning and shearing)

Common questions

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

Further reading: Strang, Linear Algebra and Its Applications; Kreyszig, Advanced Engineering Mathematics (linear algebra).

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