## Definition Of Matrix.

### The content of the lecture.

These are the headline points for the content of the next posts for matrices.

1-Definitions of matrices.

2-Matrices properties.

3-How to use Gaussian elimination.

4-Crammer’s rule.

### What is a matrix?

What are the matrices? the first definition of the matrix, the matrix is a rectangular array of numbers bounded by brackets.

If we have, for instance, two intersecting lines. The first line is represented by the equation x+2y=+3, while the second line is represented by the equation x-y=-3.

By using the algebraic method, we can get the point of intersection, which will be(-1,+2).

This point will satisfy the equations of the two lines. The X and y values will satisfy the two equations.

### How do we get to the intersecting point by using matrices?

For this kind of problem, whether two or three lines and want to find out the point of intersection, how can we find that point using matrices?

First, we start by writing the matrix. The matrix coefficient in our case consists of two columns, the first for the coefficient of x, (1 1).

While the second column is for the coefficient of y, (-1,+2), these two columns are bounded by a bracket and are called matrix coefficient and located to the left, the second vector is called column Matrix or vector of a column matrix, then followed by the =at the right side of two equations are two numbers,( -3,3).

The A values can be written by estimating the values of a_{1},b_{1},a_{2}, and b_{2}. and will be called coefficients. The slide image shows the necessary calculations.

These numbers will be placed in a column (-3,3). This vector column is called a constant. The previous form of the matrix can be further rewritten in the form of A*X=B.

Two A is a matrix, with 2×2, which means that it has two rows and two columns and can be abbreviated as AX=B. The two lines have an intersecting point, and we can get the value of x, and y after solving these equations.

### What is the augmented matrix?

What is the augmented matrix meaning? Augmented as a translation means reinforced if we consider the column, which is at the right side of the equal sign, and add this column to the previous matrix (2×2), which has two rows and two columns.

Add a line separator as we can see, for the Augmented matrix, a new matrix is formed, it has three rows and three columns and is called the Augmented matrix. The new shape is a matrix (m x n).

To explain the (m x n )matrix, in the first row, the first element is a11, then a12, till a1n.

The first letter is for the first-row number, while the second number 1 is for the column number, so a11 is an element located at a first row -first column, the first column is from a11 till am1, while the second column is from a12 till am2, the last element is (man). The matrix, which we have has n columns.

The n can be given values from unlimited numbers. (a-mn) is the last element. m is no of rows, and n is the number of columns.

### What are the different types of matrices?

Now, we will start with some definitions, first, the square matrix, for which the number of rows= no. of columns for instance for a 2×2 matrix, the elements can be arranged as (a11,a12, a21, a22). The row matrix is the matrix in the form of a line.

For the example shown, we have ( 1×4 ), a matrix, and one bracket 1 by 4, since we have one row and 4 columns. Matrix with one row and the number of columns.

For a useful external link, and types of matrices, math is fun.

This is the next post, Types of Matrices.