Play around with different values in the matrix to see how the linear transformation it represents affects the image. Notice how the sign of the determinant (positive or negative) reflects the orientation of the image (whether it appears "mirrored" or not). The arrows denote eigenvectors corresponding to eigenvalues of the same color.

\[ \begin{bmatrix} \FormInput[2][matrix-entry][1]{a} & \FormInput[2][matrix-entry][0]{b}\\ \FormInput[2][matrix-entry][0]{c} & \FormInput[2][matrix-entry][1]{d} \end{bmatrix} \]

Determinant: \(\)

Eigenvalues: \(\), \(\).

Change image:

Here are some examples of matrix transformations.

Transformation Matrix Try it
Rotation by angle \(\theta\) \(\begin{bmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{bmatrix}\) \(\theta\):
Reflection about line at angle \(\theta\) \(\begin{bmatrix}\cos2\theta & \sin2\theta\\\sin2\theta & -\cos2\theta\end{bmatrix}\) \(\theta\):
Shear parallel to \(x\)-axis \(\begin{bmatrix}1 & k\\0 & 1\end{bmatrix}\) \(k\):
Shear parallel to \(y\)-axis \(\begin{bmatrix}1 & 0\\k & 1\end{bmatrix}\) \(k\):
Uniform scaling by factor \(c\) \(\begin{bmatrix}c & 0\\0 & c\end{bmatrix}\) \(c\):