Math for Machine Learning - Chapter 9: Determinants

This quiz rigorously tests your understanding of determinants, focusing on their role in linear transformations and implications for machine learning. It delves into the geometric interpretation of determinants as measures of space scaling and orientation, the critical concept of singularity, and its direct link to matrix invertibility. Questions will challenge your knowledge of advanced determinant properties, their application to various matrix types (orthogonal, similar, block matrices), and their connection to eigenvalues and system solvability. Mastery of these concepts is crucial for understanding topics like PCA, linear regression, and optimization in ML.

Key formulas and concepts covered:

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

• Cofactor expansion for $n \times n$ matrices: $\text{det}(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}$ (for expansion along row $i$, where $M_{ij}$ is the determinant of the submatrix formed by removing row $i$ and column $j$).

• Product rule: $\text{det}(AB) = \text{det}(A) \text{det}(B)$

• Inverse rule: $\text{det}(A^{-1}) = \frac{1}{\text{det}(A)}$ (if $A$ is invertible)

• Transpose rule: $\text{det}(A^T) = \text{det}(A)$

• Scalar multiplication: $\text{det}(kA) = k^n \text{det}(A)$ for an $n \times n$ matrix $A$.

• Singularity: A matrix $A$ is singular if and only if $\text{det}(A) = 0$. This implies columns (or rows) are linearly dependent, and the transformation collapses space.

• Invertibility: A matrix $A$ is invertible if and only if $\text{det}(A) \neq 0$. An invertible matrix represents a transformation that can be undone.

• Geometric Interpretation: $|\text{det}(A)|$ represents the scaling factor of volume (or area in 2D) under the transformation represented by $A$. The sign of $\text{det}(A)$ indicates whether the transformation preserves ($+$) or reverses ($-$) orientation.

• Eigenvalues: The product of eigenvalues of a matrix $A$ equals its determinant: $\text{det}(A) = \prod_{i=1}^{n} \lambda_i$.