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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:

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