Lecture 006

3 \times 3-Determinant: \det \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i\\ \end{vmatrix} = a \det \begin{vmatrix} e & f\\ h & i\\ \end{vmatrix} - b \det \begin{vmatrix} d & f\\ g & i\\ \end{vmatrix} + c \det \begin{vmatrix} d & e\\ g & h\\ \end{vmatrix}

Submatrix notation: A_{ij} is the submatrix after remove i-th row and j-th column. (i, j)-minor is \det A_{ij}.

n \times n-Determinant:

Properties of Determinant:

Orthogonal Matrix: a square matrix such that columns (and rows) are orthogonal and unit length.

For Q \in \mathbb{R}^{n \times n}, the followings are equivalent

Determinant rank: \text{det-rank}(A) = \max(\{k | B^{k \times k} = \text{submatrix}(A) \land \det B \neq 0\})

Cramer's rule: \{\frac{\det A_i(b)}{\det A} | 1 \leq i \leq n\} is the unique solution to any system Ax = b if A is invertible // TODO: proof

Cofactor and Adjoint:

Geometric Volume:

Geometric Colinear

Table of Content