# j-james/math

All my math notes, now in Markdown.

View the Project on GitHub j-james/math

# Matrices

## Learning Targets

You should be able to

• Know how to algebraically add / subtract / multiply matrices
• Understand and apply inverse operations with matrices
• Operations and solving with a graphing calculator

## Concepts / Definitions

### Definition

Let $m$ and $n$ be positive integers. An $m$ by $n$ matrix is a rectangular array of $m$ rows by $n$ columns of real numbers.

### Elements of a Matrix

$m$ = number of rows
$n$ = number of columns
$a_{mn}$ = any element in row $m$ and column $n$
$m \times n$ = dimension of a matrix

$A = \begin{bmatrix} a_{11}&a_{12}&\dots&a_{1n}\\ a_{21}&a_{22}&\dots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\dots&a_{mn}\\ \end{bmatrix}$

### Operations

Add / subtract corresponding entries, same size matrix.

#### Scalar Multiplication

Multiply each entry by the scalar number, same size matrix.

#### Matrix Multiplication (NOT commutative)

Multiply each row of first by each column of second; need to correspond accordingly and each matrix and answer may not be same size

$"Dot\ Product"$ $\begin{bmatrix} \color{red}{1}&\color{red}{2}&\color{red}{3}\\ 4&5&6\\ \end{bmatrix} \times \begin{bmatrix} \color{blue}{7}&8\\ \color{blue}{9}&10\\ \color{blue}{11}&12\\ \end{bmatrix} = \begin{bmatrix} \color{purple}{58}&\ \\ \ &\ \\ \end{bmatrix}$ $(1)(7) + (2)(9) + (3)(11) = 58$

### Identity Matrix

$I_1 = [1] ,\ I_2 = \begin{bmatrix} 1&0\\ 0&1\\ \end{bmatrix} ,\ I_3 = \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix} ,\ I_n = \begin{bmatrix} 1&0&0&\dots&0\\ 0&1&0&\dots&0\\ \vdots&\vdots&\vdots&\ddots&\dots\\ 0&0&0&\dots&1\\ \end{bmatrix}$

$$AB \ = \begin{bmatrix} \ \ \ 3&\ \ \ 4\\ -1&-1\\ \end{bmatrix} \begin{bmatrix} -1&-4\\ \ \ \ 1&\ \ \ 3\\ \end{bmatrix}$$ $$\qquad = \begin{bmatrix} (3*-1)+(4*1)&(3*-4)+(4*3)\\ (-1*-1)+(-1*1)&(-1*-4)+(-1*3)\\ \end{bmatrix}$$ $$\qquad = \begin{bmatrix} -3+4&-12+12\\ 1+-1&4+-3\\ \end{bmatrix}$$ $$\qquad = \begin{bmatrix} 1&0\\ 0&1\\ \end{bmatrix}$$

### Determinants for 2x2 and 3x3

$A = \begin{bmatrix} a&b\\ c&d\\ \end{bmatrix} \quad \lvert A\rvert = ad - bc$

$$\begin{vmatrix} a&b&c\\ d&e&f\\ g&h&i\\ \end{vmatrix} \; =\ a \begin{vmatrix} e&f\\ h&i\\ \end{vmatrix} -\ b \begin{vmatrix} d&f\\ g&i\\ \end{vmatrix} +\ c \begin{vmatrix} d&e\\ g&h\\ \end{vmatrix}$$ $$\,\quad\qquad\qquad\qquad = a(ei-fh) - b(di-fg) + c(dh-eg)$$ $$\quad\qquad\qquad\qquad = aei - afh - bdi - + bfg + cdh - ceg$$ $$\!\qquad\qquad\qquad\qquad = (aei + bfg + cdh) - (afh + bdi + ceg)$$

### Inverse Matrices

$A^{-1} = \begin{bmatrix} a&b\\ c&d\\ \end{bmatrix}^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d&-b\\ -c&a\\ \end{bmatrix}$ $\$ $A^{-1} = \frac{1}{\lvert A\rvert} \begin{bmatrix} \begin{vmatrix} a_{22}&&a_{23}\\ a_{32}&&a_{33}\\ \end{vmatrix}& \begin{vmatrix} a_{13}&&a_{12}\\ a_{33}&&a_{32}\\ \end{vmatrix}& \begin{vmatrix} a_{12}&&a_{13}\\ a_{22}&&a_{23}\\ \end{vmatrix}\\ \begin{vmatrix} a_{23}&&a_{21}\\ a_{33}&&a_{31}\\ \end{vmatrix}_{\;}& \begin{vmatrix} a_{11}&&a_{13}\\ a_{31}&&a_{33}\\ \end{vmatrix}_{\;}& \begin{vmatrix} a_{13}&&a_{11}\\ a_{23}&&a_{21}\\ \end{vmatrix}_{\;}\\ \begin{vmatrix} a_{21}&&a_{22}\\ a_{31}&&a_{31}\\ \end{vmatrix}& \begin{vmatrix} a_{12}&&a_{11}\\ a_{32}&&a_{31}\\ \end{vmatrix}& \begin{vmatrix} a_{11}&&a_{12}\\ a_{21}&&a_{22}\\ \end{vmatrix}\\ \end{bmatrix}$ $\$ $A x A^{-1} = I \implies \begin{bmatrix} 1&2&3\\ 4&5&6\\ 7&8&9\\ \end{bmatrix} \times \begin{bmatrix} \;&\;&\;\\ \;&\;&\;\\ \;&\;&\;\\ \end{bmatrix} = \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix}$

### Solving a Matrix Equation

$AX = B \quad\qquad\qquad$ Given; since A is $n \times n$, $X$ must by $n \times p$
$A^{-1}(AX) = A^{-1}B :\;\;$ Multiply on the left by $A^{-1}$
$(A^{-1}A)X = A^{-1}B :\;\;$ Associative property of matrices
$(I_n)X = A^{-1}B :\qquad$ Property of matrix inverses
$X = A^{-1}B \,:\;\;\quad\qquad$ Property of the identity matrix

#### Solving Example

$$a + b + c = -5$$ $$9a + 3b + c = 5$$ $$16a + 4b + c = 16$$

$[A] \begin{bmatrix} a\\ b\\ c\\ \end{bmatrix} = [B] \implies \begin{bmatrix} a\\ b\\ a\\ \end{bmatrix} = [A]^{-1}[B]$ $\begin{matrix} a+b+c=-5\\ 9a+3b+c=5\\ 16a+4b+c=16\\ \end{matrix} \implies \begin{bmatrix} 1&1&1\\ 9&3&1\\ 16&4&1\\ \end{bmatrix} \begin{bmatrix} a\\ b\\ c\\ \end{bmatrix} = \begin{bmatrix} -5\\ 5\\ 16\\ \end{bmatrix}$

(use calculator to solve)