02 Matrices

A matrix is an array of symbols consisting of $m$ rows and $n$ columns. Matrices are usually given a capital letter for their symbol.

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

Size

The size of a matrix is its number of rows by its number of columns. The size of A would be $m \times n.$

Example: if $ B = \begin{bmatrix} 2 & -7 & 5 \\ 0 & 3 & 1 \end{bmatrix},$ then $B$ is a $2\times 3$ matrix.

Entries

Entry: A single symbol in a matrix.

For example, using the same matrix $B$, we can refer to entries using the following format: $b_{ij},$ which would point to the entry at the $i$th row and $j$th column of $B.$

$$ \begin{aligned} b_{23} & = 1 \\ b_{32} &\text{ undefined.} \end{aligned} $$


Elementary Row Operations

There are 3 operations:

  1. Replacement: Replace a row with the sum of said row and a multiple of another row.

$$ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} R_{2}-2R_{1} \equiv \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 0 \\ 7 & 8 & 9 \end{bmatrix} $$

  1. Interchange: Swap 2 rows.

$$ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} R_{3} \leftrightarrow R_{1} \equiv \begin{bmatrix} 7 & 8 & 9 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \end{bmatrix} $$

  1. Scale: Multiply a row by a nonzero number.

$$ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} R_{1}\cdot 3 \equiv \begin{bmatrix} 3 & 6 & 9 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} $$