Linear Algebra - Matrices: (lesson 1 of 3)

## Matrices Definitions

### Square matrix:

If a matrix A has n rows and n columns then it can be said that it's a square
matrix.

Example 1:

$
A = \left( {\begin{array}{*{20}{c}}
1&2&3\\
4&5&6\\
7&8&9
\end{array}} \right)
$

### Diagonal matrix

A diagonal matrix is a square matrix with all non-diagonal elements
being 0. The diagonal matrix is completely denoted by the diagonal elements.

Example 2:

$
A = \left( {\begin{array}{*{20}{c}}
1&0&0\\
0&5&0\\
0&0&9
\end{array}} \right)
$

The matrix is denoted by the diagonal $(1 , 5 , 9)$

### Row matrix

A matrix with one row is called the row matrix

Example 3:

$
A = \left[ {\begin{array}{*{20}{c}}
1&2&3
\end{array}} \right]
$

### Column matrix

A matrix with one column is called the column matrix

Example 4:

$
A = \left[ {\begin{array}{*{20}{c}}
1\\
4\\
7
\end{array}} \right]
$

### Matrices of the same kind

Matrix $A$ and $B$ are of the same kind if $A$ has as many rows and as many columns as $B$

### The transpose of a matrix

The $n \times m$ matrix $A_T$ is the transpose of the $m \times n$ matrix $A$ if and only
if the ith row of $A$ is equal to the ith
column of $A_T$ for $(i = 1,2,3,..n)$.

Example 5:

$
A = \left[ {\begin{array}{*{20}{l}}
{\color{red}{1}}&{\color{red}{2}}&{\color{red}{3}}\\
4&5&6
\end{array}} \right] \to {A^T} = \left[ {\begin{array}{*{20}{l}}
{\color{red}{1}}&4\\
{\color{red}{2}}&5\\
{\color{red}{3}}&6
\end{array}} \right]
$

### 0-matrix

When all the elements of a matrix $A$ are 0, we call it the 0-matrix.

Example 6:

$
A = \left[ {\begin{array}{*{20}{c}}
0&0&0\\
0&0&0\\
0&0&0
\end{array}} \right]
$

### An identity matrix $I$

An identity matrix $I$ is a diagonal matrix with all diagonal element equal to 1

Example 7:

$
A = \left[ {\begin{array}{*{20}{c}}
1&0&0\\
0&1&0\\
0&0&1
\end{array}} \right]
$

### The opposite matrix of a matrix

If the sign of all the elements of a matrix $A$ are changed, that matrix is the
opposite matrix $-A$.

Example 8:

$
A = \left( {\begin{array}{*{20}{c}}
1&{ - 2}&3\\
{ - 4}&{ - 5}&{ - 6}\\
{ - 7}&8&9
\end{array}} \right) \to A' = \left( {\begin{array}{*{20}{c}}
{ - 1}&2&{ - 3}\\
4&5&6\\
7&{ - 8}&{ - 9}
\end{array}} \right)
$

### A symmetric matrix

A square matrix is called symmetric if it is equal to its transpose.

Example 9:

$
A = \left( {\begin{array}{*{20}{c}}
1&2&3\\
2&5&6\\
3&6&9
\end{array}} \right)
$