Solve the system of linear equations using Gauss-Jordan elimination. x+2 y-z= 6 2

Solve the system of linear equations using Gauss-Jordan...

Key Concepts

System of Linear Equations

A system of linear equations is a set of equations where each equation is linear in the variables involved. The solutions to these systems are the values of the variables that satisfy all equations simultaneously, and such systems frequently occur in various fields of mathematics, physics, and engineering.

Matrix Representation

Matrix representation involves expressing a system of linear equations as an augmented matrix, where the coefficients of the variables form the main part of the matrix and the constants form the augmented column. This form simplifies the handling and manipulation of the system, particularly for algorithmic methods like elimination processes.

Elementary Row Operations

Elementary row operations are the basic manipulations allowed on the rows of a matrix to facilitate solving the corresponding system of equations. These include swapping two rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another. These operations are crucial because they transform the matrix into a simpler form without changing the solution set.

Gauss-Jordan Elimination

Gauss-Jordan elimination is an algorithmic method used to solve systems of linear equations by converting the augmented matrix into a simpler form, typically the reduced row echelon form. This method extends Gaussian elimination by further eliminating nonzero entries above the pivots, resulting in a unique solution or identifying when there are infinitely many or no solutions.

Reduced Row Echelon Form (RREF)

The reduced row echelon form of a matrix is a canonical form achieved after applying elementary row operations. In this form, each pivot (the first nonzero element from the left in a nonzero row) is 1, and it is the only nonzero entry in its column. RREF provides a straightforward way to read the solutions of the system or determine if the system is inconsistent.

Transcript

00:01 So we're going to solve this system of equations using gauss -jordan elimination.

00:06 So right here i have an augmented matrix with each entry in the matrix corresponding to the coefficients.

00:15 So like in the first row, it's 1, 2, negative 1, and 6.

00:20 And right here i have 1 -2 -9 -1 -6 for the first equation coefficients.

00:27 Okay.

00:27 Okay, so what we want to do now is we want to get this in row -reduced echelon form.

00:34 So it's going to take a couple steps.

00:37 First we want to get rid of the two and the three in the first column.

00:41 So let's multiply the first row by negative two, and we'll add that with the second row.

00:59 So then we get 0, negative 5, 5, and negative 25.

01:09 And then let's multiply the first row.

01:10 Row by negative 3 and add it with the third row.

01:15 So negative 3, negative 6, 3, and negative 18.

01:26 And then we have 3, negative 2, 3, negative 16.

01:33 So that'll be 0, negative 8, 6, and negative 18, minus 16, which is negative 34.

01:47 So now let me rewrite this.

01:55 So we have 1, 2, negative 1, 6 for a matrix, and then this row is now 0, negative 5, 5, negative 25.

02:07 So i'm actually going to multiply the second row by negative, or i'm going to divide the second row by negative 5.

02:16 So what that will give me is 1, negative 1, and then this will be 5.

02:30 Okay, now this last row will be 0, negative 8, 6, negative 34.

02:42 Okay, next we want to get this negative 8 in the third column.

02:46 Out of here we want it to be a 0, so we're going to multiply the second row by 8.

02:51 So that will be 0, 8, negative 8, 40, and then that third row, we're going to add it to the third row.

03:10 And then it'll be 0, 0, negative 2, and 6.

03:18 Okay.

03:20 Now let's rewrite that third row.

03:27 So 1, 2, negative 1, 6, 0, 1, negative 1, 5...

Please give Ace some feedback