Introduction to Matrices

INTRO

A matrix is simply an array of numbers, and we usually denote it by a capital letter.

\[ \text{e.g}. \quad A = \begin{bmatrix}
5 & 1 & 7 \\
2 & -1 & 3
\end{bmatrix} \\\]
In general, we would have \(A =
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{12} & \cdots & a_{2n} \\
\vdots & \vdots & & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\

\end{bmatrix} \). Notice that we refer to entries as \(a_{ij}\) where \(i\) is the row and \(j\) is the column of the entry we are talking about. The size of the matrix is \(m \times n\) where \(m\) is the number of rows and \(n\) is the number of columns.

Example: If \(
A = \begin{bmatrix}
5 & 1 & 7 \\
2 & -1 & 3
\end{bmatrix}
\), find \(a_{21}\) , \(a_{13}\), and determine the size of the matrix.

Solution:
There are 2 rows and three columns so the size of the matrix is 2 x 3. Also, \(a_{21} = 2\) because this refers to the entry in the 2nd row and the 1st column. Finally, \(a_{13} = 7\) because this refers to the entry in the 1st row and the 3rd column.