Understand Rectangular Matrices to Row Echelon Form

Hi friends, in today’s class we all are going to discuss about one of the most interesting topic of mathematics that is Rectangular Matrices To Row Echelon Form, algebra 2 for Grade X of gujarat board. To solve many problem in Linear Algebra require that a matrix be converted into one of two forms that is Row Echelon Form and the Reduced Row Echelon Form. Matrix is a rectangular array of real numbers(Real Numbers Definition) with m rows and n columns. If a matrix has m rows and n columns then it is called a matrix of order m x n and a matrix with an equal number of rows and column is called square matrix. If the number of rows and columns are same then it is said to have order n. The elements from the upper left to the lower right of the matrix are the main diagonal of a square matrix. The row matrix is a matrix that has only one row and column matrix is a matrix that has only one column. If a matrix has only one row or one column then it is called a vector. The standard form of matrix is given below.To get its detailed explanation click here,

                                  1      4      5

            A  =              6     3      2

                                5      2       1
From the above it is a m x n order matrix.
Now we are converting Linear Equations to Matrices of a system. When we are converting Linear Equations to Matrices then each equation in the system becomes a row and each variable in the system becomes a column. The coefficients are placed into a matrix and variables are dropped. If the right hand side is included then it is called an augmented matrix and if the right hand side is not included then it is called a coefficient matrix. For example the system of linear equations is
6x + y – 3z = 1
3x – y + 2z = 3
4x + 7y – z = 9

And becomes the augmented matrix like      x       y       z     rhs

                                                                              6     1        -3     1

                                                                              3    -1         2     3

                                                                              4     7         -1    9

A rectangular matrix is said to be in row-echelon form if its augmented matrix is in row echelon form and meet the following conditions.

All zero rows belong at the bottom of the matrix or in other words If there is a row of all zeros, and then it is at the bottom of the matrix. If first non-zero element of any row is a 1 then that element is called the leading one. The leading one of a row is in a column to the right of the leading one of the row above it and one more thing the row space of the row echelon form is the same as that of the original matrix. The example of row-echelon form is given below.

                           1     1     4     4

                          0      1     0     3

                          0      0     0     1

The above matrix is 3 x 4 orders. Let’s we take another example

                           1     0     0

                           0     1     0

                          0      0     1

In above form of matrix every leading coefficient is 1 and is the only nonzero entry in its column.

This is all for today. In next class I am going to discuss about other topics of matrix and to know more about Matrix and Vector Multiplication in grade X 
and Statistical experiments in Grade XI then wait till next session.