Write the coefficient matrix and the augmented matrix for each system. { x+z+w =-1 x+y+2 w

Write the coefficient matrix and the augmented matrix for...

Key Concepts

Augmented Matrix

The augmented matrix combines the coefficient matrix with the constant terms from the right-hand side of the equations into a single matrix. This matrix is used to perform row operations in order to solve the system, providing a compact and systematic way to apply techniques like Gaussian elimination or Gauss-Jordan elimination to find solutions.

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. In the context of linear algebra, these systems are typically analyzed to determine the values of the variables that satisfy all equations simultaneously, and they can be represented in various forms including matrix form which simplifies computations and theoretical analysis.

Coefficient Matrix

The coefficient matrix is a matrix constructed from the coefficients of the variables in the system of equations, with each row corresponding to an equation and each column corresponding to a variable. This matrix encapsulates the structure of the system without considering the constant terms, and it is essential in methods such as Gaussian elimination, determinant calculations, and other linear transformations.

Transcript

00:01 So the question wants us to figure out what the coefficient matrix will be for this system of equations as well as the augmented matrix.

00:08 So to figure out what the coefficient matrix is going to be, we can take the rows of this coefficient matrix to be each of these equations here.

00:23 And the columns will be assorted with the variables.

00:31 So the columns will be, the first column can be for the x variable, the second column can be for the y variable, third column can be for the z variable, and the last column can be for the w variable.

00:45 So let's first look at this first column here, or the x column.

00:53 So we're going to go through each of these equations here and figure out what the coefficient is in front of the x for each of these terms in each of these equations.

01:06 Once again, because this is the coefficient, the coefficient matrix, we don't need to include these constants here at the end that don't have any variables attached to them.

01:18 All right, so for the first equation, the coefficient in front of the x variable is 1.

01:25 For the second equation, it's 1 again.

01:29 For the third equation, there is no x term, so it's 0.

01:33 And for the last equation, it's 2.

01:37 For the y column, in the first equation, it's going to be equal to 0, because there is no y term.

01:45 In the second equation, it's equal to 1...

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