# Matrices

Definition: A Matrix $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$ is a rectangular array of numbers or expressions known as entries to the matrix. If $A$ has $m$ rows and $n$ columns then we say that the size of $A$ is $m \times n$. |

The following examples represents some matrices of varying sizes.

(1)We will now look at some other important definitions.

Definition: A matrix $A$ is said to be Square if it has the same number of rows as it does columns, that is if $A$ is of size $m \times n$ then $m = n$. A general square matrix is commonly denoted with size $n \times n$. |

Definition: If $A$ is an $m \times n$ matrix, the entry in row $i$ and column $j$ (clearly $i, j$ are integers and $1 ≤ i ≤ m$ and $1 ≤ j ≤ n$) is denoted by $a_{ij}$ (e.g. $a_{23}$ represents the entry in row $2$ and column $3$ of a matrix $A$). |

Consider the following matrix $B = \begin{bmatrix} 4 & 1\\ 2 & 7 \end{bmatrix}$. This matrix is square since it has the same number of rows ($2$) as it does columns ($2$), and $B$ has entries $b_{11} = 4$, $b_{12} = 1$, $b_{21} = 2$ and $b_{22} = 7$.

We will now look at some elementary row operations on matrices that we will eventually utilize to help us with solving systems of linear equations.

# Elementary Row Operations

There are 3 elementary row operations that we can perform on matrices. The notation in brackets indicate the notation for the operation applied where $a$ and $b$ represent any arbitrary row in the matrix to which the operations are applied.

**1.**Multiplying a row by a constant $k$ where $k ≠ 0$ ($kR_{a} \to R_{a}$).

**2.**Adding (or subtracting) a multiple $k$ of a row to another ($R_a + kR_b \to R_a$).

**3.**Interchanging two rows ($R_a \leftrightarrow R_b$).

For example, consider the following matrix $\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}$.

Let's now multiply the first row by the constant $3$, that is perform the row operation $3R_{1} \to R_{1}$. We thus obtain the matrix $\begin{bmatrix} 3 & 6\\ 3 & 4 \end{bmatrix}$.

Now let's take row $1$ and add twice row $2$ to it, that is $R_1 + 2R_2 \to R_1$. We obtain $\begin{bmatrix} 9 & 14 \\ 3 & 4 \end{bmatrix}$.

Finally, let's interchange rows $1$ and $2$ as $R_{1} \leftrightarrow R_{2}$ to get $\begin{bmatrix} 3 & 4\\ 9 & 14 \end{bmatrix}$.

Definition: A set of matrices are said to be Row Equivalent if a series of finite elementary row operations can be performed in order to obtain one matrix from the next. |

In our example above, all of the matrices we listed were considered row equivalent as we can easily obtain one from another with just a few of these elementary row operations.