Matrices and Elementary Row Operations

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)

\begin{align} A_{2 \times 3} = \begin{bmatrix} 2 & -2 & 4\\ 1 & 0 & \pi \end{bmatrix} \quad , \quad B_{4\times1} = \begin{bmatrix} 5 \\ x\\ 2\\ x^2 \end{bmatrix} \quad , \quad C_{3 \times 3} = \begin{bmatrix} x^2 & 0 & sin(x)\\ 2 + x & -1 & \frac{1}{2}\\ (x+3)^2 & x & 0.4 \end{bmatrix} \end{align}

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.