3Blue1Brown: The Determinant - Chapter 6 Quiz

The determinant: area, volume and orientation

This quiz follows Chapter 6 of 3Blue1Brown’s Essence of Linear Algebra. It combines geometric interpretation with small determinant calculations.

Practice

• Compute 2×2 determinants and interpret their sign and magnitude.
• Distinguish signed area or volume from ordinary nonnegative measurements.
• Recognize dimension collapse and dependent matrix columns.
• Reason about transformed unit squares and unit cubes.
• Combine determinant effects of successive transformations.

For a square matrix $$\begin{pmatrix}a&b\c&d\end{pmatrix}$$, the determinant is $$ad-bc$$. Its absolute value scales ordinary area. In three dimensions, the absolute determinant scales ordinary volume. A negative nonzero determinant reverses orientation; zero determinant indicates loss of dimension.

Prior knowledge of vector coordinates, basis images and matrix-vector multiplication is useful. Several questions require multiple selections. Check each statement separately, and consider the zero matrix as well as nondegenerate examples. When using parallelogram vertices, identify the ordered basis images before determining a sign. Work through calculations and sketches before reading the explanations.