SYSTEMS OF EQUATIONS in THREE VARIABLES
It is often desirable or even necessary to use more than one variable to model a
situation in a field such as business, science, psychology, engineering, education, and
sociology, to name a few. When this is the cas
Computation of Eigenvectors
Let A be a square matrix of order n and
eigenvector of A associated to
one of its eigenvalues. Let X be an
. We must have
This is a linear system for which the matrix coefficient is
. Since the zerovector is a solution, the sys
Determinants of Matrices of Higher Order
As we said before, the idea is to assume that previous properties satisfied by the
determinant of matrices of order 2, are still valid in general. In other words, we
assume:
1.
Any matrix A and its transpose have t
Determinant and Inverse of Matrices
Finding the inverse of a matrix is very important in many areas of science. For
example, decrypting a coded message uses the inverse of a matrix. Determinant may
be used to answer this problem. Indeed, letA be a square
Computation of Eigenvalues
For a square matrix A of order n, the number
exists a non-zero vector C such that
is an eigenvalue if and only if there
Using the matrix multiplication properties, we obtain
This is a linear system for which the matrix coefficie
Eigenvalues and Eigenvectors: An
Introduction
The eigenvalue problem is a problem of considerable theoretical interest and wideranging application. For example, this problem is crucial in solving systems of
differential equations, analyzing population gro
IntroductiontoDeterminants
For any square matrix of order 2, we have found a necessary and sufficient condition
for invertibility. Indeed, consider the matrix
The matrix A is invertible if and only if
. We called this number
the determinant of A. It is cl
Systems of Linear Equations: Gaussian
Elimination
It is quite hard to solve non-linear systems of equations, while linear systems are quite
easy to study. There are numerical techniques which help to approximate nonlinear
systems with linear ones in the h
The Case of Complex Eigenvalues
First let us convince ourselves that there exist matrices with complex eigenvalues.
Example. Consider the matrix
The characteristic equation is given by
This quadratic equation has complex roots given by
Therefore the matri
SYSTEMS OF EQUATIONS in TWO VARIABLES
A system of equations is a collection of two or more equations with the same set of
unknowns. In solving a system of equations, we try to find values for each of the
unknowns that will satisfy every equation in the sy
System of Equations: An Introduction
Many books on linear algebra will introduce matrices via systems of linear equations.
We tried a different approach. We hope this way you will appreciate matrices as a
powerful tool useful not only to solve linear syst
Application of Determinant to Systems:
Cramer's Rule
We have seen that determinant may be useful in finding the inverse of a nonsingular
matrix. We can use these findings in solving linear systems for which the matrix
coefficient is nonsingular (or invert