MAT 350: Row Reduction Workshop

Dr. Gilbert

August 8, 2025

Reminders and Today’s Goal

  • 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.

    • Limits: Once per week until the Thanksgiving Break and then twice per week afterwards
  • 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.

Completed Examples: Example #1

Example: Reduce the matrix below to its corresponding reduced row echeclon form.

\[A = \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix}\]

\[\begin{align} \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix} \end{align}\]

Completed Examples: Example #1

Example: Reduce the matrix below to its corresponding reduced row echeclon form.

\[A = \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix}\]

\[\begin{align} \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix} &\stackrel{R_2 \leftarrow R_2 + 3R_1}{\longrightarrow} \begin{bmatrix} 1 & 4\\ 0 & 20\end{bmatrix} \end{align}\]

Completed Examples: Example #1

Example: Reduce the matrix below to its corresponding reduced row echeclon form.

\[A = \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix}\]

\[\begin{align} \begin{bmatrix} 1 & 4\\ -3 & 8\end{bmatrix} &\stackrel{R_2 \leftarrow R_2 + 3R_1}{\longrightarrow} \begin{bmatrix} 1 & 4\\ 0 & 20\end{bmatrix}\\ &\stackrel{R_2 \leftarrow \frac{1}{20}R_2}{\longrightarrow} \begin{bmatrix} 1 & 4\\ 0 & 1\end{bmatrix} \end{align}\]

Completed Examples: Example #1

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.

Completed Examples: Example #2

Example: \(C = \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} \end{align}\]

Completed Examples: Example #2

Example: \(C = \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} &\stackrel{R_2 \leftarrow R_2 + (-2)R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} \end{align}\]

Completed Examples: Example #2

Example: \(C = \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 4 & 6 & 8 & 10\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} &\stackrel{R_2 \leftarrow R_2 + (-2)R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & -1 & -2 \end{bmatrix} \\ &\stackrel{R_3 \leftrightarrow R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 0 & 1 & 0 & -1 & -2\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \end{align}\]

Completed Examples: Example #2

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.

Completed Examples: Example #3

Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} \end{align}\]

Completed Examples: Example #3

Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\stackrel{R_1 \leftarrow \frac{1}{2} R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} \end{align}\]

Completed Examples: Example #3

Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\stackrel{R_1 \leftarrow \frac{1}{2} R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} \\ &\stackrel{R_2 \leftarrow R_2 + (-1)R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 3 & 9 & 1 \end{bmatrix} \end{align}\]

Completed Examples: Example #3

Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\stackrel{R_1 \leftarrow \frac{1}{2} R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} \\ &\stackrel{R_2 \leftarrow R_2 + (-1)R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 3 & 9 & 1 \end{bmatrix} \\ &\stackrel{R_3 \leftarrow R_3 + (-3)R_1}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix} \end{align}\]

Completed Examples: Example #3

Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\sim \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix} \end{align}\]

Completed Examples: Example #3

Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\sim \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow R_3 + (-3)R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & -2 \end{bmatrix} \end{align}\]

Completed Examples: Example #3

Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\sim \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow R_3 + (-3)R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & -2 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow \left(-\frac{1}{2}\right) R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & 1 \end{bmatrix} \end{align}\]

Completed Examples: Example #3

Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\sim \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow R_3 + (-3)R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & -2 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow \left(-\frac{1}{2}\right) R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & 1 \end{bmatrix} \end{align}\]

\[\begin{align} &\stackrel{R_2 \leftarrow R_2 + (-2)R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \end{align}\]

Completed Examples: Example #3

Example: \(B = \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1\end{bmatrix}\)

\[\begin{align} \begin{bmatrix} 2 & 4 & -2\\ 1 & 3 & 1\\ 3 & 9 & 1 \end{bmatrix} &\sim \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 3 & 4 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow R_3 + (-3)R_2}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & -2 \end{bmatrix}\\ &\stackrel{R_3 \leftarrow \left(-\frac{1}{2}\right) R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 2\\ 0 & 0 & 1 \end{bmatrix} \end{align}\]

\[\begin{align} &\stackrel{R_2 \leftarrow R_2 + (-2)R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\\ &\stackrel{R_1 \leftarrow R_1 + R_3}{\longrightarrow} \begin{bmatrix} 1 & 2 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \end{align}\]

Completed Examples: Example #3

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.

Examples to Try

  • The following slides include examples for you to try on your own.
  • The groups are loosely arranged by difficulty of the row reduction tasks.
  • For each of the presented matrices, perform row reduction operations to convert the matrix into its equivalent reduced row echelon form.

Examples to Try: Group #1

  1. \(A_1 = \begin{bmatrix}1 & 2\\ 0 & 3\end{bmatrix}\)
  2. \(A_2 = \begin{bmatrix} 1 & 0\\ 0 & 1\\ 2 & 3\end{bmatrix}\)
  3. \(A_3 = \begin{bmatrix} 1 & 2 & 3\\ 0 & 1 & 4\end{bmatrix}\)

Examples to Try: Group #2

  1. \(B_1 = \begin{bmatrix} 2 & 4 & -2\\ -1 & -2 & 1\\ 1 & 1 & 0\end{bmatrix}\)
  2. \(B_2 = \begin{bmatrix} 1 & 2 & 1 & 0\\ 2 & 4 & 3 & 1\\ 1 & 1 & 2 & 3\end{bmatrix}\)
  3. \(B_3 = \begin{bmatrix} 1 & 2 & 3\\ 0 & 1 & 4\\ 0 & 0 & 1\\ 0 & 0 & 0\end{bmatrix}\)

Examples to Try: Group #3

  1. \(C_1 = \begin{bmatrix} 0 & 2 & 4 & 6\\ 1 & 3 & 1 & 5\\ 2 & 6 & 2 & 10\\ 3 & 1 & 1 & 4\end{bmatrix}\)
  2. \(C_2 = \begin{bmatrix} 1 & 1 & 1\\ 2 & 2 & 2\\ 1 & 0 & 1\\ 0 & 1 & 2\\ 3 & 3 & 3\end{bmatrix}\)
  3. \(C_3 = \begin{bmatrix} 1 & 2 & 0 & 3 & 1\\ 2 & 4 & -1 & 6 & 2\\ -1 & -2 & 1 & -3 & 0\\ 0 & 0 & 2 & 1 & 3\end{bmatrix}\)

Homework




\[\Huge{\text{Finish Homework 3}}\] \[\Huge{\text{on MyOpenMath}}\]

Next Time…




\(\Huge{\text{Vectors and Linear Combinations}}\)