# Introduction to Matrices

By **Catalin David**

A matrix is a rectangular table consisting of rows and columns containing numbers.

The general form of a matrix is:

Any element of a matrix is defined using the notation $a_{n,m}$, where m represents the row and n represents the column of the entry.

Example 1

$A=
\begin{pmatrix}
1 & 0 & 2\\
3 & 1 & 4\\
\end{pmatrix}
$

A is a matrix with 2 rows and 3 columns in which '2' is found on the first row and the third column.

Example 2

$B=
\begin{pmatrix}
1 & 5\\
2 & 8\\
7 & 3\\
\end{pmatrix}$

B is a matrix with 3 rows and 2 columns in which '8' is found on the second row and the second column.

A matrix having the same number of rows and columns is called a **square matrix**.

Example 3 $C= \begin{pmatrix} 1 & 2 & 3\\ 3 & 7 & 2\\ 4 & 5 & 1\\ \end{pmatrix}$

C is a matrix with 3 rows and 3 columns.

D is the general form of a square matrix.

$D= \begin{pmatrix} \color{red}{a_{1,1}} & a_{1,2} & a_{1,3} & . & . & \color{blue}{a_{1,n}}\\ a_{2,1} & \color{red}{a_{2,2}} & a_{2,3} & . & \color{blue}{a_{2,n-1}} & a_{2,n}\\ a_{3,1} & a_{3,2} & \color{red}{a_{3, \color{blue}{3}}} & . & . & a_{3,n}\\ . & \color{blue}{a_{n-1,2}} & . & . & .& .\\ \color{blue}{a_{n,1}} & a_{n,2} & a_{n,3} & . & . & \color{red}{a_{n,n}}\\ \end{pmatrix}$

The main diagonal is represented by the red entries, while the secondary diagonal is represented by the blue entries.

A square matrix with all elements found on the main diagonal equal to 1 and the others equal to 0 is called the identity matrix.

Example 4

$I_{2}=
\begin{pmatrix}
1 & 0\\
0 & 1\\
\end{pmatrix}$

Example 5

$I_{3}=
\begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
\end{pmatrix}$

**The transpose of a matrix** is obtained by switching the rows with the columns in the initial matrix. If A is the given matrix, its transpose is $A^{T}$.

Example 6

$A=\begin{pmatrix} 1 & 3\\ 5 & 9 \end{pmatrix}$ | thus | $A^{T}=\begin{pmatrix} 1 & 5\\ 3 & 9 \end{pmatrix}$ |