Activity 1.3.2

Enter the following matrix into Sage. $$\left[ \begin{array}{rrrr} -1 & -2 & 2 & -1 \\ 2 & 4 & -1 & 5 \\ 1 & 2 & 0 & 3 \end{array} \right]$$

Give the matrix the name $A$ by entering

A = matrix( ..., ..., [ ... ])
	      

We may then find its reduced row echelon form by entering

A = matrix( ..., ..., [ ... ])
A.rref()
	      

A common mistake is to forget the parentheses after rref.

Use Sage to find the reduced row echelon form of the matrix from this item of this activity.

Use Sage to describe the solution space of the system of linear equations $$\begin{alignedat}{5} -x_1 & & & & & {}+{} & 2x_4 & {}={} & 4 \\ & & 3x_2 & {}+{} & x_3 & {}+{} & 2x_4 & {}={} & 3 \\ 4x_1 & {}-{} & 3x_2 & & & {}+{} & x_4 & {}={} & 14 \\ & & 2x_2 & {}+{} & 2x_3 & {}+{} & x_4 & {}={} & 1 \\ \end{alignedat}$$

Consider the two matrices: $$\begin{array}{rcl} A & = & \left[ \begin{array}{rrrr} 1 & -2 & 1 & -3 \\ -2 & 4 & 1 & 1 \\ -4 & 8 & -1 & 7 \\ \end{array}\right] \\ B & = & \left[ \begin{array}{rrrrrr} 1 & -2 & 1 & -3 & 0 & 3 \\ -2 & 4 & 1 & 1 & 1 & -1 \\ -4 & 8 & -1 & 7 & 3 & 2 \\ \end{array}\right] \\ \end{array}$$ We say that $B$ is an augmentation of $A$ because it is obtained from $A$ by adding some more columns.

Using Sage, define the matrices and compare their reduced row echelon forms. What do you notice about the relationship between the two reduced row echelon forms?

Using the system of equations in this item, write the augmented matrix corresponding to the system of equations. What did you find for the reduced row echelon form of the augmented matrix?

Now write the coefficient matrix of this system of equations. What does this item of this activity tell you about its reduced row echelon form?