# Matrices

Matrix: A matrix is a rectangular array of numbers or expressions arranged in rows and columns.

• Equality: same size and entries are equal
• Sum: add all entries together (only defined on similarly sized matrices)

## Matrix multiplication #

Matrix multiplication/product is only defined when the number of columns of the first matrix is equal to the number of rows of the second matrix: $$(m \times n) \cdot (n \times k) = (m \times k)$$

Every (i, j)-entry in the resulting matrix:

$$(AB)_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \dots + a_{in}b_{nj}$$

Matrix multiplication is also associative $A(BC)=(AB)C$.

Identity matrix $I_n$: ones on diagonal, otherwise 0. $$I_3 = \begin{pmatrix} 1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1 \end{pmatrix}$$

Any matrix multiplied by the identity matrix is itself.

## Matrix powers #

• $A^k = A \dots A$ ($k$ times)
• $A^0 = I_n$ (identity matrix)
• Transpose $A^T$: columns and rows are swapped

## Matrix inverses #

A $n \times n$ matrix is invertible/nonsingular: if there exists an $n \times n$ matrix $C$ such that: $$CA = I ~\land~ AA^{-1} = I$$

Otherwise $A$ is called not invertible / singular

Inverse of a $2 \times 2$ matrix: $$A = \begin{pmatrix} a & b\\\ c & d \end{pmatrix}$$

$$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\\ -c & a \end{pmatrix}$$

Note: $\det A = ad = bc$

The $n \times n$ matrix $A$ is invertible if and only if:

• The determinant of $A$ is $not$ zero
• The number $0$ is not an eigenvalue of $A$

Algorithm for finding $A^{-1}$:

1. Row reduce the augmented matrix: $[~A~~I~]$
2. If $A$ is row equavalent to $I$ then $[~A~~I~] = [~I~~A^{-1}~]$ . Otherwise $A$ does not have an inverse.