September 24, 2025
Example: Consider the following augmented coefficient matrices in their reduced row echelon form (RREF). In each case, determine the number of solutions to the underlying linear system, write those solutions using parametric vector form, and describe the geometry of the solution space (point, line, plane, etc.).
\[i)~~~\left[\begin{array}{rr|r} 1 & 0 & 2\\ 0 & 1 & -1\\ 0 & 0 & 1\end{array}\right]~~~~~~ii)~~~\left[\begin{array}{rrr|r} 1 & 0 & 4 & 7\\ 0 & 1 & -2 & 1\end{array}\right]~~~~~~iii)~~~\left[\begin{array}{rrrrr|r} 1 & -3 & 4 & 0 & 2 & 3\\ 0 & 0 & 0 & 1 & 6 & -4\\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\]
\[iv)~~~\left[\begin{array}{rrrr|r} 1 & 0 & 0 & 0 & -3\\ 0 & 1 & 0 & 0 & 5\\ 0 & 0 & 1 & 0 & 8\\ 0 & 0 & 0 & 1 & -1\end{array}\right]~~~~~~v)~~~\left[\begin{array}{rrrr|r} 1 & 0 & 2 & -1 & 4\\ 0 & 1 & 3 & 5 & -2\end{array}\right]\]
The purpose of today’s class meeting is simple – build confidence with row reduction.
As mentioned previously, you’ll have a row reduction Gateway Exam in MAT350 worth 15% of your overall course grade that you’ll need to pass before the end of the semester.
You’ll be able to retake the Gateway Exam multiple times to show mastery.
Each Gateway Exam will consist of four matrices for you to reduce within a strict 20-minute time limit.
You’ll need to correctly reduce at least three of the matrices to their reduced row echelon form in order to pass.
First Attempt: Your first attempt at the Gateway Exam will take place at the end of this class meeting. Prior to that, we’ll take time to practice.
Example: Reduce the matrix below to its corresponding reduced row echeclon form.
\[A = \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix}\]
\[\begin{align} \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix} \end{align}\]
Example: Reduce the matrix below to its corresponding reduced row echeclon form.
\[A = \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix}\]
\[\begin{align} \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix} &\stackrel{R_2 \leftarrow R_2 + 3R_1}{\longrightarrow} \begin{bmatrix} 1 & 4\\ 0 & 20\end{bmatrix} \end{align}\]
Example: Reduce the matrix below to its corresponding reduced row echeclon form.
\[A = \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix}\]
\[\begin{align} \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix} &\stackrel{R_2 \leftarrow R_2 + 3R_1}{\longrightarrow} \begin{bmatrix} 1 & 4\\ 0 & 20\end{bmatrix}\\ &\stackrel{R_2 \leftarrow \frac{1}{20}R_2}{\longrightarrow} \begin{bmatrix} 1 & 4\\ 0 & 1\end{bmatrix} \end{align}\]
Example: Reduce the matrix below to its corresponding reduced row echeclon form.
\[A = \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix}\]
\[\begin{align} \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix} &\stackrel{R_2 \leftarrow R_2 + 3R_1}{\longrightarrow} \begin{bmatrix} 1 & 4\\ 0 & 20\end{bmatrix}\\ &\stackrel{R_2 \leftarrow \frac{1}{20}R_2}{\longrightarrow} \begin{bmatrix} 1 & 4\\ 0 & 1\end{bmatrix}\\ &\stackrel{R_1 \leftarrow R_1 + (-4)R_2}{\longrightarrow}\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}~~\blacktriangledown \end{align}\]
This was a square matrix with a pivot in every row and every column.
Example: \(C = \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} \end{align}\]
Example: \(C = \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} &\stackrel{R_2 \leftarrow R_2 + (-2)R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} \end{align}\]
Example: \(C = \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} &\stackrel{R_2 \leftarrow R_2 + (-2)R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} \\ &\stackrel{R_3 \leftrightarrow R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 0 & 1 & 0 & -1 & -2\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \end{align}\]
Example: \(C = \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} &\stackrel{R_2 \leftarrow R_2 + (-2)R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} \\ &\stackrel{R_3 \leftrightarrow R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 0 & 1 & 0 & -1 & -2\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \\ &\stackrel{R_1 \leftarrow R_1 + (-2)R_2}{\longrightarrow} \begin{bmatrix} 1 & 0 & 3 & 6 & 9\\ 0 & 1 & 0 & -1 & -2\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} ~~\blacktriangledown \end{align}\]
This was a rectangular matrix with a pivots missing in both rows and columns.
Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} \end{align}\]
Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\stackrel{R_1 \leftarrow \frac{1}{2} R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} \end{align}\]
Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\stackrel{R_1 \leftarrow \frac{1}{2} R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} \\ &\stackrel{R_2 \leftarrow R_2 + (-1)R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 3 & 9 & 1 \end{bmatrix} \end{align}\]
Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\stackrel{R_1 \leftarrow \frac{1}{2} R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} \\ &\stackrel{R_2 \leftarrow R_2 + (-1)R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 3 & 9 & 1 \end{bmatrix} \\ &\stackrel{R_3 \leftarrow R_3 + (-3)R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix} \end{align}\]
Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\sim \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix} \end{align}\]
Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\sim \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow R_3 + (-3)R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & -2 \end{bmatrix} \end{align}\]
Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\sim \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow R_3 + (-3)R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & -2 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow \left(-\frac{1}{2}\right) R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & 1 \end{bmatrix} \end{align}\]
Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\sim \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow R_3 + (-3)R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & -2 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow \left(-\frac{1}{2}\right) R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & 1 \end{bmatrix} \end{align}\]
\[\begin{align} &\stackrel{R_2 \leftarrow R_2 + (-2)R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \end{align}\]
Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\sim \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow R_3 + (-3)R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & -2 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow \left(-\frac{1}{2}\right) R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & 1 \end{bmatrix} \end{align}\]
\[\begin{align} &\stackrel{R_2 \leftarrow R_2 + (-2)R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\\ &\stackrel{R_1 \leftarrow R_1 + R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \end{align}\]
Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)
\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\sim \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow R_3 + (-3)R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & -2 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow \left(-\frac{1}{2}\right) R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & 1 \end{bmatrix} \end{align}\]
\[\begin{align} &\stackrel{R_2 \leftarrow R_2 + (-2)R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\\ &\stackrel{R_1 \leftarrow R_1 + R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \\ &\stackrel{R_1 \leftarrow R_1 + (-2)R_2}{\longrightarrow} \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}~~\blacktriangledown \end{align}\]
Another square matrix with pivots in every row and column.
\[\Huge{\text{Finish Homework 3}}\] \[\Huge{\text{on MyOpenMath}}\]
\(\Huge{\text{Vectors and Linear Combinations}}\)