3Blue1Brown: The Determinant - Chapter 6 Quiz

This quiz rigorously tests your understanding of the determinant concept as presented in the 3Blue1Brown 'Essence of Linear Algebra' Chapter 6 video. It covers the determinant's role as an area/volume scaling factor, its implications for orientation reversal, the meaning of a zero determinant (dimension collapse, linear dependence), the geometric intuition behind the 2x2 formula ($ad-bc$), the extension to 3D with parallelepipeds and the right-hand rule, and the property $\text{det}(M_1 M_2) = \text{det}(M_1) \text{det}(M_2)$. The questions range from conceptual interpretations to application-based scenarios, requiring deep insight into the visual and geometric aspects of linear transformations.

Key Formulas and Concepts:

• 2D Determinant: For a matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\text{det}(A) = ad - bc$.

• Geometric Meaning (2D): $\text{det}(A)$ is the signed area of the parallelogram formed by the transformed basis vectors $\text{L}(\hat{i}) = \begin{pmatrix} a \\ c \end{pmatrix}$ and $\text{L}(\hat{j}) = \begin{pmatrix} b \\ d \end{pmatrix}$. The absolute value $|\text{det}(A)|$ is the area scaling factor. A negative sign indicates orientation inversion.

• Geometric Meaning (3D): $\text{det}(A)$ is the signed volume of the parallelepiped formed by the transformed basis vectors $\text{L}(\hat{i})$, $\text{L}(\hat{j})$, $\text{L}(\hat{k})$. The absolute value $|\text{det}(A)|$ is the volume scaling factor. A negative sign indicates orientation inversion (e.g., changing from right-hand rule to left-hand rule).

• Zero Determinant: If $\text{det}(A) = 0$, the transformation collapses space into a lower dimension (e.g., a line or a point in 2D, a plane, line, or point in 3D). This implies the column vectors of $A$ are linearly dependent.

• Determinant of Product: For matrices $M_1$ and $M_2$, $\text{det}(M_1 M_2) = \text{det}(M_1) \text{det}(M_2)$. This reflects the cumulative effect of sequential scaling factors.

• Basis Vectors: The transformation of the unit square (2D) or unit cube (3D), whose edges are defined by the standard basis vectors, provides the fundamental geometric interpretation of the determinant.