The determinant is the signed area of the parallelogram spanned by i and j. A negative determinant mirrors the shape, and zero flattens it onto a line.
The determinant is positive. The F reads forwards.
i1.00, 0.00
j0.00, 1.00
Area1.00
The area of the image is the absolute value of the determinant. A negative sign mirrors the F. Zero lays the F on a line.
2×2 matrix. The columns are i and j. Read each row from left to right.
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Where this is used
The absolute value of the determinant tells how much area or volume grows or shrinks. In physics and engineering, the same factor is called the Jacobian.
A technical artist checks the sign. A negative determinant means the model was mirrored, and the surface normals can face the wrong way.
In chemistry, a related idea is chirality. Some mirror-image molecules are not the same as the original.
A determinant of zero is a warning. The shape has fallen onto a line or a plane. A set of equations may then have no single answer.
Common mistakes
The sign matters. Reading only the absolute value misses the mirror.
A reflection has determinant −1, not 1. The F reads backwards.
A determinant of zero means the shape has fallen onto a line. That matrix has no inverse.