Determinant [2x2]

The determinant of a matrix is a helpful tool for multiple applications. One of them is for example computing the eigenvalues of a matrix.

The determinant succeeds in characterizing certain properties of a matrix and the linear transformation represented by the matrix. The determinant can only be found for matrices that are square. Meaning that the matrix needs to have the same amount of rows and columns and thus has dimensions n-by-n.

The determinant is always a scalar value and this value is dependent on the matrix entries, which are the values inside the matrix. Now suppose that we have a square matrix A of dimension 2x2.

The determinant of this matrix $\mathbf{A}$ can be computed in the following way

Visualization det(2x2)

If the matrix $\mathbf{A}$ is considered to be a transformation matrix this determinant can be visualized as the area within the white parallelogram of the figure.

Let's work out the exact area of the white parallelogram which is the result after a transformation caused by $\mathbf{A}$.

First, the area of the complete square needs to be computed and equals

If the other green parts are now subtracted from the total area we get the area within the parallelogram (in short par)

Reordering the terms gives

And finally there can be seen that most parts cancel each other out

The remaining part comes down to $ad-cb$, which is exactly the same result as the determinant calculated above in the first section. Therefore, the determinant of a 2-by-2 matrix is actually the area of the parallelogram resulting from the linear transformation.