1. What is an augmented matrix used for?

An augmented matrix is used to represent a system of linear equations. This is different from the coefficient matrix because it not only includes the coefficients but the constants as well. (Renee DeCarlo)

2. List the elementary row operations. Why are they important?

The elementary row operations are:

1. Interchange two rows

2. Multiply a row by a nonzero constant

3. Add a multiple of a row to another row

They are important in producing equivalent systems in order to help in solving a system of linear equations. (Greg Kehoe)

Instructor’s comment: The goal is to produce an equivalent system that is simpler to solve than the original.

3. Why is it not possible for a homogeneous system of equations to be inconsistent?

In order to be consistent, the system has to have at least 1 solution. Every homogeneous system will be consistent because of the trivial solution. No matter what, you will always get a solution by setting each variable equal to 0. Then, both the left-hand and right-hand sides of each equation will equal 0. 0 = 0, and therefore it is impossible for a homogeneous system to be inconsistent. (Jeff Stark)

The goal of polynomial fitting is to take a collection of points and find a polynomial function that passes through these points. (Karla Helstrom)

Two matrices (A and B) can be added only if the number of rows in A equals the number of rows in B and the number of columns in A equals the number of columns in B. They must be the same size. (Bonnie Parker)

The matrices do not have to be of the same size, but the number of columns of the first matrix must equal the number of rows of the second matrix. (Jessica Murray)

These matrix operations may also be used to express a system of linear equations in another equivalent form. This new expression for the system is a matrix equation. To get one specific one, multiply the coefficient matrix of the system times a column matrix of the variables and set the product equal to the column matrix of the constants the equations were equal to in the original system. To see their the same just do the matrix algebra and use matrix properties. (Adam Weisblatt)

AB=BA sometimes. An example where they are not equal is the matrices A=[1, 3; 2, -1] and B=[2, -1; 0, 2] where AB=[2, 5; 4, -4] and BA=[0, 7; 4, -2].

An example of where they are equal is the matrices A=[1, 2; 1, 1] and B=[-2, 4; 2, -2] where AB=[-2, 4; 4, -2] and BA=[-2, 4; 4, -2]. (Abby Moyer)

An identity matrix is a special type of square matrix that has 1's on the main diagonal and 0's elsewhere. some of its properties are that if A is a matrix of size m x n then AIn= A and ImA=A. (Katherine Hobart)

The transpose of a matrix is simply thaking a matrix and switching the rows to columns and the columns to rows. ie.) [1 2; 3 4] transposed is [1 3; 2 4] (Tim Jenny)