Triangular Forms for Linear Systems Gaussian Elimination Elementary Row Operations and Row Echelon Form Reduced Row Echelon Form Solving Systems with Inverse Matrices Applications … and why Many applications in business and science are modeled by systems of linear equations in three or more variables.
The following operations produce an equivalent system of linear equations. 1. Interchange any two equations of the system. 2. Multiply (or divide) one of the equations by any nonzero real number. 3. Add a multiple of one equation to any other equation in the system. Transforming a system to triangular form is called Gaussian elimination.
Solve the system of equations using Gaussian elimination. Mult. 1 st equation by -2 and add to 2 nd equation, replacing the 2 nd. Mult. 1 st equation by -4 and add to 3 rd equation, replacing the 3 rd. Mult. 2 nd equation by -1 and add to 3 rd equation, replacing the 3 rd. Triangular form makes the solution easy to read.
Mult. 1 st equation by -2 and add to 2 nd equation, replacing the 2 nd. Mult. 1 st equation by 3 and add to 3 rd equation, replacing the 3 rd. Mult. 2 nd equation by 2 and add to 3 rd equation, replacing the 3 rd.
A matrix is in row echelon form if the following conditions are satisfied. 1. Rows consisting entirely of 0’s (if there are any) occur at the bottom of the matrix. 2. The first entry in any row with nonzero entries is 1. 3. The column subscript of the leading 1 entries increases as the row subscript increases. Another way to phrase parts 2 and 3 is to say that the leading 1’s move to the right as we move down the rows.
A combination of the following operations will transform a matrix to row echelon form. 1. Interchange any two rows. 2. Multiply all elements of a row by a nonzero real number. 3. Add a multiple of one row to any other row.
The augmented matrix of this system of equations is:
Indicates interchanging the ith and jth row of the matrix. Indicates multiplying the ith row by −2.
Indicates multiplying the ith row by −3 and adding it to the jth row.
If we continue to apply elementary row operations to a row echelon form of a matrix, we can obtain a matrix in which every column that has a leading 1 has 0’s elsewhere. This is the reduced echelon form.