Matrices and Determinants
Source lecture(s): PHY621 Lec1
Intuition
Matrices compress many simultaneous linear equations into one compact object. The determinant tells you whether the map is invertible.
Formal Definition
An \(N\times N\) matrix \(\mathbf{A}\) maps \(\vec{x}\mapsto\vec{b}=A\vec{x}\). The determinant \(\det\mathbf{A}\) scales volume under this map.
Mathematical Formulation
\[\det\mathbf{A}=\sum_{\sigma}\text{sgn}(\sigma)\prod_{i=1}^N A_{i,\sigma(i)}\]
Derivation
The determinant is the unique multilinear, alternating \(N\)-linear function with \(\det\mathbf{I}=1\). Laplace expansion computes it from any row or column.
Worked Example
\(A=\begin{pmatrix}1&2\\3&4\end{pmatrix}\) has \(\det A = 1\cdot4-2\cdot3=-2\).
Common Mistakes
- Multiplying diagonal entries for non-triangular matrices.
- Confusing \(\det(AB)\) with \(\det A+\det B\).
Related Concepts
Quiz
Q1. When is a matrix singular?
Answer
When its determinant is zero.
Q2. Does \(\det(A^T)=\det A\)?
Answer
Yes.