Matrices

Matrices

A matrix is a rectangular arrangement of numbers. In a matrix you will find numers in rows and columns that express linear system equations.

Source: http://www.catonmat.net/blog/wp-content/uploads/2009/11/matrix-form-two-equations.gif

If we want to solve systems of equations we must consider that:

We want to find values of x and y that make both equations true!

We can make following row transformations: 

1. Interchange rows.

2. Multiply all elements of any row by the same number (nonzero).

3. Add to all elements of a row a multiple of the corresponding elements of another row.

To solve a linear equation system we do row transformations till we obtein a matrix in its triangular form, like:

1 a b c d

0 1 e f g

0 0 1 h i

0 0 0 1 j

0 0 0 0 1

We have different types of martices, check this out:

We have a tool to know if a matrix has solutions or not: the determinant.
Lets see how to obtain the determinant of a matrix:
if we have two equations:
ax+by=c
dx+ey=f
The determinant will be:

Note that we have two lines, and the slopes will be -a/b and -d/e, for this reason if the determinant is zero, we know the lines are identicals because the slopes are equal! What does it means? We have an infinite number of solutions! 

In the other hand, if the determinant is nonzero we have different lines and there is an intersect in one point.

Example:



The determinant for this case will be: (2)(2)-(-1)(-1)=3
We know this system have a solution.