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the chapter on matrices is very instructive and requires some concentration on the part of the student. This course will allow him to develop new skills.

## I. The notion of matrix.

Let m, n and p be three non-zero natural numbers.

### 1. notion of matrix and operations.

A matrix of size (or format) is an array of real numbers n rows and p columns.

Example:

is a matrix with 2 rows and 3 columns and therefore of size .

When n = 1, we say that M is a **row matrix**, formed by a single row.

Then, when p = 1, we say that M is a **column matrix**, formed by a single column.

And when n = p, we say that M is a **square matrix** of order n.

A **diagonal matrix** is a square matrix, whose terms are all zero except

when i=j.

The **identity matrix** of order n is the diagonal matrix of order n whose coefficients

diagonals are equal to 1. It is noted .

The null matrix of size , denoted , is the matrix of size , whose coefficients are all zero.

Examples:

Two matrices A and B of size are **equal** when, for any and

we have .

A square matrix of order is **symmetric** when, for any and

we have .

Example:

The following matrix is symmetric.

### 2. Operations on matrices.

Let and be two matrices of size .

- The
**sum of the matrices A and B**, noted A + B, is the matrix of size

such that, for all and, we . - The product of the matrix A by a real , noted , is the matrix of

size such that, for all and , we have .

Let A, B and C be three matrices of the same size and and two real numbers.

- A+ B = B + A (commutativity of the sum of matrices)
- A+(B +C) = (A+ B)+C (associativity of the sum of matrices)

We call the matrix , noted , the opposite matrix of A such that,

for all and , we have .

Furthermore, we note the matrix .

Examples:

Let be a row matrix of size and

a column matrix of size column matrix of size .

Then the product is the real number defined by :

Example:

### 3.product of two matrices.

If A is a matrix of size and B a matrix of size , the **product of**

**matrices** A and B, noted or , is the matrix of size such that,

for all and , we have .

In other words, the element . is the product of the i-th line of A by the j-th

B column.

Properties:

Let A, B and C be three matrices and a real number.

Subject to the definition of products and sums, we have :

Let A be a square matrix of order n and k a nonzero natural number.

The **k-th power** of A, denoted , is the matrix (k times ).

### 4.invert of a matrix and system resolution.

A square matrix A of size n is **invertible** when there is a square matrix B

of size n such that .

Let be a square matrix of order 2.

The determinant of A is the real, denoted det(A), defined by .

A square matrix is **invertible** if, and only if, its **determinant is non-zero**.

In particular, if
is invertible then

Let A be a square matrix of size n and X and B two column matrices with n rows.

If A is invertible, then the matrix writing system AX = B admits a unique

solution given by the column matrix .

Example:

## II. Graphs.

A **graph** is a representation composed of **vertices** (points) connected by

edges (segments).

A **directed graph** is a graph whose edges have a direction of travel.

The**order** of a graph is the number of vertices of this graph.

The **degree of** a vertex is the number of edges incident to this vertex, without taking into account

their possible direction of travel.

Example:

The graph below is of order 5.

The vertices K and L are of degree 3.

The vertices M , M and M, are of degree 2.

Two vertices are **adjacent** when they are connected by at least one edge.

A graph is **complete** when all its vertices are adjacent to each other.

Examples:

1. The graph below is complete because all its vertices are two by two adjacent.

2. The graph below is not complete because the vertices A and B, for example, are not

not adjacent.

For an undirected graph, a **chain** is a sequence of consecutive edges connecting

two vertices (possibly merged).

The **length of a chain** is the number of edges in it.

For a directed graph, a **path** is a sequence of consecutive edges connecting two

vertices (possibly merged) taking into account the direction of travel of the edges.

## III. Application of matrix calculus to graphs.

Let n be a non-zero natural number. Consider a graph of order n (directed or not)

whose vertices are numbered from 1 to n, then arranged in ascending order.

The **adjacency matrix** of this graph is the square matrix of size n, whose

coefficient is equal to the number of edges starting from vertex i to arrive at vertex j.

Example:

By noting M the adjacency matrix of the graph below obtained by arranging

the vertices in alphabetical order.

We have:

Let n and k be two non-zero natural numbers and M the adjacency matrix of a graph

of order n, whose vertices are numbered from 1 to n and arranged in ascending order

The term of the i-th row and the j-th column of the matrix gives the

number of chains (or paths) of length k connecting vertex i to vertex j.

Example:

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