Math for Machine Learning - Chapter 7: Matrix Operations and Properties

This quiz is designed for an advanced assessment of your understanding of matrix operations and their properties, a cornerstone of machine learning. It delves deep into multiplication, transpose, and inverse, focusing not just on computation but on the profound geometric interpretations and abstract properties that are critical for developing and analyzing ML models. Expect questions that challenge your knowledge of how transformations affect geometric space, the conditions for matrix invertibility, the nuances of matrix multiplication commutativity, and the properties of special matrices like orthogonal, symmetric, and projection matrices. Success in this quiz indicates a masterful grasp of the linear algebra that underpins algorithms from linear regression to deep learning.

Key Formulas:

• Matrix Multiplication: If $A$ is $m \times n$ and $B$ is $n \times p$, then $C = AB$ is an $m \times p$ matrix where $C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$ - Non-commutativity: In general, $AB \neq BA$ - Transpose Properties: $(A^T)_{ij} = A_{ji}$, $(AB)^T = B^T A^T$, $(A+B)^T = A^T + B^T$ - Inverse Properties: For an invertible square matrix $A$, $AA^{-1} = A^{-1}A = I$. The inverse exists iff $\det(A) \neq 0$. Also, $(AB)^{-1} = B^{-1}A^{-1}$ and $(A^T)^{-1} = (A^{-1})^T$.