INTRODUCTION TO MATRIX

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A matrix is a rectangular array of numbers. 

In a matrix, the set of numbers aligned horizontally are called rows and the sets of numbers aligned vertically are called columns.

Each number of a matrix is called an entry.

The size or dimensions of a matrix are usually defined as $m \times n$, with m indicating rows and n indicating column

For example, the matrix:

$\left[{\begin{array}{cccc} 6 & 3 & 7 \\ 7 & 8 & 6 \\ \end {array} } \right]$ is a 2 by 3 matrix.

Please note that matrix entries are first defined by rows and then by columns. There are different types of matrices. Let look at some of them.

A square matrix is a matrix that has the same numbers of rows and columns.

For example, 2 by 2 matrix, 3 by 3 matrix are all square matrices.

$\left[{\begin{array}{cc} 4 & 6 \\ 5 & 7  \\ \end {array} } \right]$ is a 2 by 2 matrix

$\left[{\begin{array}{cccc} 6 & 3 & 7 \\ 7 & 8 & 6 \\  9 & 2 & 1 \\ \end {array} } \right]$ is a 3 by 3 matrix

A row matrix is a matrix consisting of one row. It usually has the dimension $1 \times n$ 

For example, $\left[{\begin{array}{cc} 4 & 6 & 7  \\ \end {array} } \right]$ is a row matrix because it has one row.

On the other hand, a column matrix is a matrix consisting of one column with dimensions $ m \times 1$. 

For example, $\left[{\begin{array}{cc} 4 \\ 6 \\ 7  \\ \end {array} } \right]$ is a column matrix because it has one column.

Example 1

Find the dimension of the 

$$B=\left[{\begin{array}{cccc} 6 & 3 & 7 \\ 7 & 8 & 6 \\  9 & 2 & 1 \\ \end {array} } \right]$$

Solution:

The dimension of a matrix is the number of rows by the number of columns of the matrix.

Therefore, the dimension of $B=\left[{\begin{array}{cccc} 6 & 3 & 7 \\ 7 & 8 & 6 \\  9 & 2 & 1 \\ \end {array} } \right]$ is 3 by 3

Example 2

Given that $Y=\left[{\begin{array}{cccc} 6 & 3 & 7 \\ 7 & 4 & 5 \\  6 & 2 & 1 \\ \end {array} } \right]$,

1. What are the dimension of Matrix Y

2. What is the entry in $y_{32}$

3. What is the entry in $y_{13}$

4. What is the entry in $y_{11}$

Solution:

1. The dimension of matrix Y is 3 by 3 because the matrix has 3 rows and 3 columns.

2. Entry $y_{32}$ is simply the entry at row 3, column 2, which is 2

3. Entry $y_{13}$ is simply the entry at row 1, column 3, which is 7

4. Entry $y_{11}$ is the entry at row 1, column 1 which is 6

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