#### Objective

1. To perform operations on the matrices.
2. To reduce to echelon form a matrix.
3. To recognize and calculate the rank of a matrix.
4. To solve a linear system by the Gaussian elimination method.
5. To calculate the determinant of a square matrix.
6. To recognize an invertible matrix and to calculate its inverse by various methods.
7. To solve a linear system with invertible matrix AX = B by X = A–1 B and by Cramer’s method
8. To perform the various operations on the polynomials.
9. To perform the Euclidean division of a polynomial by another.
10. To recognize a root a of a polynomial and to factorize this last by X – a.
11. To recognize a multiple root of one polynomial using the successive derivatives of polynomials.
12. To know the fundamental theorem of the algebra.
13. To factorise a real polynomial into product of polynomials of degree 1 and polynomials of degree 2 with negative discriminant.

#### Content

Complex numbers : Reminders – De Moivre formula - nth roots a complex number - nth roots of the unit.
Matrices : Definitions – Operations - Examples of matrices - Elementary Operations – Echelon form - Rank of a matrix - Transposed of a matrix.
Invertible matrices - Calculation of the inverse of a matrix by the method of the elementary operations.
Linear systems : Definitions – Examples - Augmented matrix - Classification of the linear systems - Resolution of a linear system by Gaussian elimination method - Resolution of a system using invertible matrix and by Cramer’s method - Application to the calculation of the solution of homogeneous linear systems.
Determinants : Definitions - Properties - Application to the calculation of the inverse of a matrix and to the resolution of the linear systems with invertible matrices.
Polynomials : Definitions – Operations - Examples of Polynomials - Euclidean Division – G.C.D. of a polynomial - Derivative of Polynomials – Taylor’s formula - Roots (simple and multiples) - fundamental Theorem of the algebra - Factorisation of a Polynomial in R [X].
Comment : This course is a course of calculus. It is addressed to the students in requires to avoid the two theoretical approaches.
Knowledge of the students as regards complex numbers is not homogeneous what requires a detailed attention. It is necessary to avoid building rigorously the field of the complex numbers, moreover the students do not know what it is a field. One will be able to leave the algebraic form of the complex numbers that the majority of the students know. One will insist on the exponential and trigonometric forms. One will stress the de Moivre formula and its trigonometric applications. As regards nth roots of a complex number and the unit, it is necessary to insist on the calculation of these roots and their geometrical representation.
One will clarify the particular role of the nth root = e2i/n by showing that nth roots of unit are powers of this element and that the points which represent these nth roots on the trigonometrical circle are obtained starting from the point which represents by successive rotations of center the origin and angle 2i/n . One will insist in particular on the nth roots of the unit for n = 2,3 and 4.
A matrix will be defined like a table of real or complex numbers. One will take care that the student controls the elementary operations which constitute the basic tool in this course. One will characterize the invertible matrices by the row and the determinant.
Out of matter of linear system, one will take care that the student controls their resolution by echelon form, without however being unaware of that, in certain particular cases, it is possible to proceed differently. One will insist, for the homogeneous systems, with what studying it manages to determine the principal unknown factors and the secondary unknown factors after the echelon form of the system.