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:
first row: \det A = \sum_{j = 1}^n (-1)^{j + 1} a_{ij} \det A_{ij}
along i-th row: \det A = \sum_{j = 1}^n (-1)^{i+j} a_{ij} \det A_ij
along j-th column: \det A = \sum_{i = 1}^n (-1)^{j + j} a_{ij} \det A_{ij}
\det I_n = 1
matrix of all 1: \begin{cases} 0 & \text{if } n \geq 2\\ 1 & \text{if } n = 1 \text{ (single element)}\\ \end{cases}
permutation matrix: \pm 1
Properties of Determinant:
if A has a row/column of all 0, then \det A = 0.
by swapping two rows / columns, the determinant gets multiplied by -1.
if A has two identical row/column, then \det A = 0.
multiply one row/column by k, then \det A \rightarrow k\det A.
if for some rows A_i, B_i, C_i in matrix A, B, C such that A_i + B_i = C_i (and everything else are the same), then \det A + \det B = \det C. (same is true for columns)
Multiplying a row by k results determinant multiply by k
Adding a multiple of a row/column to another row/column doesn't change the determinant.
gaussian process does not change determinant. (except swaping two rows changes the sign)
\det A \neq 0 \iff A \text{ is invertible}
\det A = \det A^T
\det (AB) = \det A \cdot \det B // TODO: proof
If A invertible, then \det A^{-1} = \frac{1}{\det A}
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
A \text{ is orthogonal}
Transpose Identity: Q Q^T = I
Transformation preserve length of vector: (\forall x \in \mathbb{R}^n)(\|Qx\| = \|x\|)
Transformation preserve relative angle of vector: (\forall x, y \in \mathbb{R}^n)(Qx \cdot Qy = x \cdot y)
Determinant rank: \text{det-rank}(A) = \max(\{k | B^{k \times k} = \text{submatrix}(A) \land \det B \neq 0\})
Theorem: for any A \in \mathbb{R}^{n \times n}, \text{row-rank}(A) = \text{column-rank}(A) = \text{determinant-rank}(A)
// TODO: proof
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:
A = (a_{ij})_{1 \leq i, j \leq n}
(i, j)-cofactor of A is C_{ij} = (-1)^{i+j} \det A_{ij}
Adjoint Matrix: \begin{align*} \text{adj} A = &(C_{ij})_{1 \leq i, j \leq n}^T\\ = &(C_{ji})_{1 \leq i, j \leq n}\\ = &\begin{pmatrix} C_{11} & C_{21} & ... & C_{n1}\\ C_{12} & C_{22} & ... & C_{n2}\\ ... & ... & ... & ...\\ C_{1n} & C_{2n} & ... & C_{nn}\\ \end{pmatrix} \end{align*}
Theroem: If A \in \mathbb{n \times n} is invertible, then A^{-1} = \frac{1}{\det A} \text{adj} A // TODO: proof
Geometric Volume:
In 2D: area of parallelogram spanned by 2 vectors
In 3D: volume of parallelotope spanned by 3 vectors
In nD: hyper-volume spanned by v_1, ..., v_n is \left| \det \begin{pmatrix} v_1\\ ...\\ v_n\\ \end{pmatrix}\right|
Geometric Colinear
2D colinear test: (a_1, a_2), (b_1, b_2), (c_1, c_2) are collinear iff \begin{vmatrix} a_1 & a_2 & 1\\ b_1 & b_2 & 1\\ c_1 & c_2 & 1\\ \end{vmatrix} = 0
3D colinear test: (a_1, a_2, a_3), (b_1, b_2, b_3), (c_1, c_2, c_3), (d_1, d_2, d_3) are in the same plane (coplanar) iff \begin{vmatrix} a_1 & a_2 & a_3 & 1\\ b_1 & b_2 & b_3 & 1\\ c_1 & c_2 & c_3 & 1\\ d_1 & d_2 & d_3 & 1\\ \end{vmatrix} = 0
2D line: line from (b_1, b_2), (c_1, c_2) can be expressed as \begin{vmatrix} x & y & 1\\ b_1 & b_2 & 1\\ c_1 & c_2 & 1\\ \end{vmatrix} = 0
3D line: line from (b_1, b_2, b_3), (c_1, c_2, c_3), (c_1, c_2, d_3) can be expressed as \begin{vmatrix} x & y & z & 1\\ b_1 & b_2 & b_3 & 1\\ c_1 & c_2 & c_3 & 1\\ d_1 & d_2 & d_3 & 1\\ \end{vmatrix} = 0
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