Section02 Permutations and Cofactors

• Mothods of calculating determinants: pivots, big formula, cofactors.

The Pivot Formula

• Formula without row exchange $$\det \boldsymbol{A} = (\det \boldsymbol{L})(\det \boldsymbol{U}) = (1)(d_{1}d_{2} \cdots d_{n}) = \prod_{i=1}^{n}{d_{i}}$$

• Formula with row exchange $$\begin{array}{c} (\det \boldsymbol{P})(\det \boldsymbol{A}) = (\det \boldsymbol{L})(\det \boldsymbol{U})\\ \det \boldsymbol{P} = \pm 1 \\ \Downarrow \\ \det \boldsymbol{A} = \left\{ \begin{array}{rl} \prod_{i=1}^{n}d_{i} & \text{if the times of rows exchange is even.} \\ -\prod_{i=1}^{n}d_{i} & \text{if the times of rows exchange is odd.} \end{array} \right. \end{array}$$

• Pivots from determinants $$d_{k} = \frac{\prod_{i=1}^{k}d_{i}}{\prod_{i=1}^{k-1}d_{i}} = \frac{\det \boldsymbol{A}_{k}}{\det \boldsymbol{A}_{k-1}}$$

The Big Formula for Determinants

• According to the basic rules (Last Section) $$\begin{array}{c} \text{Split each row } i \text{ to:} \\ \boldsymbol{a}_{i \boldsymbol{\cdot}} = \left[ \begin{matrix} a_{i1} & 0 & \cdots & 0 \end{matrix} \right] + \left[ \begin{matrix} 0 & a_{i2} & \cdots & 0 \end{matrix} \right] + \cdots + \left[ \begin{matrix} 0 & 0 & \cdots & a_{in} \end{matrix} \right] \\ \Downarrow \\ \begin{array}{ll} \left\vert \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right\vert & = \left\vert \begin{matrix} a_{11} & & \\ & a_{22} & \\ & & a_{33} \end{matrix} \right\vert + \left\vert \begin{matrix} & a_{12} & \\ & & a_{23} \\ a_{31} & & \end{matrix} \right\vert + \left\vert \begin{matrix} & & a_{13} \\ a_{21} & & \\ & a_{32} & \end{matrix} \right\vert \\ & + \left\vert \begin{matrix} a_{11} & & \\ & & a_{23} \\ & a_{32} & \end{matrix} \right\vert + \left\vert \begin{matrix} & a_{12} & \\ a_{21} & & \\ & & a_{33} \end{matrix} \right\vert + \left\vert \begin{matrix} & & a_{13} \\ & a_{22} & \\ a_{31} & & \end{matrix} \right\vert\\ & = a_{11}a_{22}a_{33} \left\vert \begin{matrix} 1 & & \\ & 1 & \\ & & 1 \end{matrix} \right\vert + a_{12}a_{23}a_{31} \left\vert \begin{matrix} & 1 & \\ & & 1 \\ 1 & & \end{matrix} \right\vert + a_{13}a_{21}a_{32} \left\vert \begin{matrix} & & 1 \\ 1 & & \\ & 1 & \end{matrix} \right\vert \\ & + a_{11}a_{23}a_{32} \left\vert \begin{matrix} 1 & & \\ & & 1 \\ & 1 & \end{matrix} \right\vert + a_{12}a_{21}a_{33} \left\vert \begin{matrix} & 1 & \\ 1 & & \\ & & 1 \end{matrix} \right\vert + a_{13}a_{22}a_{31} \left\vert \begin{matrix} & & 1 \\ & 1 & \\ 1 & & \end{matrix} \right\vert \end{array} \end{array}$$

• BIG FORMULA $$\begin{array}{ll} \det \boldsymbol{A} & = \text{sum over all } n! \text{column permutations } \boldsymbol{P} = (\alpha, \beta, \cdots, \omega) \\ & = \sum ({\det \boldsymbol{P}}) a_{1 \alpha}a_{2\beta} \cdots a_{n\omega} \end{array}$$

Determinants by Cofactors

• $3$ by $3$ example $$\begin{array}{c} \begin{array}{ll} \left\vert \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right\vert & = \left\vert \begin{matrix} a_{11} & & \\ & a_{22} & a_{23} \\ & a_{32} & a_{33} \end{matrix} \right\vert + \left\vert \begin{matrix} & a_{12} & \\ a_{21} & & a_{23} \\ a_{31} & & a_{33} \end{matrix} \right\vert + \left\vert \begin{matrix} & & a_{13} \\ a_{21} & a_{22} & \\ a_{31} & a_{32} & \end{matrix} \right\vert \\ & = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\\ \end{array} \\ \\ C_{1j} = (-1)^{1+j}(\det M_{1j}) \end{array}$$

• COFACTOR FORMULA $$\begin{array}{c} \det \boldsymbol{A} = a_{i1} C_{i1} + a_{i2} C_{i2} + \cdots + a_{in} C_{in}\\ C_{ij} = (-1)^{i+j}\det M_{ij} \end{array}$$

• According to $\det \boldsymbol{A}^{\text{T}} = \det \boldsymbol{A}$, Cofactor could down column $j$ : $$\det \boldsymbol{A} = a_{1j}C_{1j} + a_{2j}C_{2j} + \cdots + a_{nj}C_{nj}$$

• Cofactors are useful when matrices have many zeros. When a row of $\boldsymbol{A}$ has a lot of zeros, we only need a few cofactors.

• Anohter description: The determinants is the dot product of any row of $\boldsymbol{A}$ with its row of cofactors.

Properties from Exercises

1. When the size of Matrix is less than $3$ by $3$, the number of odd permutations is even ($\frac{n!}{2}$ is odd, when $n \le 3$)
2. When each $a_{ij}$ in $\boldsymbol{A}$ multiplied by $\frac{i}{j}$, the $\det \boldsymbol{A}$ is unchanged.
   • proof $$\begin{array}{ll} \because & \vert \boldsymbol{A} \vert = \sum \pm a_{1\alpha}a_{2\beta}\cdots a_{n\omega} \\ \therefore & \vert \boldsymbol{A}' \vert = \sum \pm \frac{\prod_{i=1}^{n}{i}}{\alpha\beta\cdots\omega} a_{1\alpha}a_{2\beta}\cdots a_{n\omega} \\ & = \sum \pm \frac{\prod_{i=1}^{n}{i}}{\prod_{i=1}^{n}{i}} a_{1\alpha}a_{2\beta}\cdots a_{n\omega} =\sum \pm a_{1\alpha}a_{2\beta}\cdots a_{n\omega} \\ & = \vert \boldsymbol{A} \vert \end{array}$$
3. $\left\vert \begin{matrix} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{0} & \boldsymbol{D} \end{matrix} \right\vert = (\det \boldsymbol{A})(\det \boldsymbol{D})$
4. $\left\vert \begin{matrix} \boldsymbol{A} & \boldsymbol{B}\\ \boldsymbol{C} & \boldsymbol{D} \end{matrix} \right\vert \color{#D0104C}{\ne} (\det \boldsymbol{AD}) - (\det \boldsymbol{BC})$
5. $\left\vert \begin{matrix} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{C} & \boldsymbol{D} \end{matrix} \right\vert= \left\vert \begin{matrix} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{0} & \boldsymbol{D}-\boldsymbol{CA}^{-1}\boldsymbol{B} \end{matrix} \right\vert = \left\vert \begin{matrix} \boldsymbol{A} \end{matrix} \right\vert \left\vert \begin{matrix} \boldsymbol{D} - \boldsymbol{CA}^{-1}\boldsymbol{B} \end{matrix} \right\vert$
   • If $\boldsymbol{AC} = \boldsymbol{CA}$ $$\left\vert \begin{matrix} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{C} & \boldsymbol{D} \end{matrix} \right\vert = \left\vert \begin{matrix} \boldsymbol{AD} - \boldsymbol{ACA}^{-1}\boldsymbol{B} \end{matrix} \right\vert = \left\vert \begin{matrix} \boldsymbol{AD} - \boldsymbol{CB} \end{matrix} \right\vert$$
6. $\left\vert \begin{matrix} \boldsymbol{0} & \boldsymbol{I} \\ \boldsymbol{I} & \boldsymbol{0} \end{matrix} \right\vert \color{#D0104C}{\ne} \left\vert \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right\vert$