Worksheets · Precalculus

Row operations worksheet

Writing a system as an augmented matrix drops the letters and keeps the numbers. The three row operations are the same three moves you already make on the equations, and reducing to an identity on the left leaves the answers on the right.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Systems and Row Operations

Date Period

Do what each question asks.

  1. Perform R1→R1+R2: [2136−1306−5−1−124]
  2. Write in reduced row-echelon form: [−22183−5−1−1246−5−18]
  3. Perform R2→R2−3R1: [−13−11−20]
  4. Write in reduced row-echelon form: [554196−1510−35655]
  5. Perform R1→R1−3R3: [−6−4−110−5−222800−5−30]
  6. Write in reduced row-echelon form: [−1−6−3−221050−42−8]
  7. Perform R2→R2−3R1: [−16−23631−34154238]
  8. Write in reduced row-echelon form: [−3−3−4−29−36−4−11−2−6−5−32]
  9. Perform R2→R2+R1: [25−310−525]
  10. Write in reduced row-echelon form: [−62−2−20−216−22111−6]
  11. Perform R2→R2+2R1: [22−2−36−42]
  12. Write in reduced row-echelon form: [−6−5−1−613331−5−3431]
  13. Perform R1→R1+R3: [−1210−6−1−138224−12]
  14. Write in reduced row-echelon form: [−5−2−629055−3503−39]
  15. Perform R1→R1+2R2: [−23−6−14351952−4−26]
  16. Write in reduced row-echelon form: [2−6026−240−18−62−511]
  17. Perform R2→R2+2R3: [4−11−1316−1333−1−11]
  18. Write in reduced row-echelon form: [−14421−4−5−6−46−3−6−6−45]
  19. Perform R2→R2+3R1: [−44122−1−4]
  20. Write in reduced row-echelon form: [35−610−1−10−4−14−51]

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Systems and Row OperationsAnswer keyVersion 1

Date Period

Do what each question asks.

  1. Perform R1→R1+R2: [2136−1306−5−1−124][14312−1306−5−1−124]
  2. Write in reduced row-echelon form: [−22183−5−1−1246−5−18][100−101010014]
  3. Perform R2→R2−3R1: [−13−11−20][−13−14−113]
  4. Write in reduced row-echelon form: [554196−1510−35655][100−301020016]
  5. Perform R1→R1−3R3: [−6−4−110−5−222800−5−30][−6−414100−5−222800−5−30]
  6. Write in reduced row-echelon form: [−1−6−3−221050−42−8][10020101001−2]
  7. Perform R2→R2−3R1: [−16−23631−34154238][−16−2366−173−6754238]
  8. Write in reduced row-echelon form: [−3−3−4−29−36−4−11−2−6−5−32][100501020012]
  9. Perform R2→R2+R1: [25−310−525][25−3120−6]
  10. Write in reduced row-echelon form: [−62−2−20−216−22111−6][1002010−6001−2]
  11. Perform R2→R2+2R1: [22−2−36−42][22−2110−46]
  12. Write in reduced row-echelon form: [−6−5−1−613331−5−3431][100−501060016]
  13. Perform R1→R1+R3: [−1210−6−1−138224−12][145−12−6−1−138224−12]
  14. Write in reduced row-echelon form: [−5−2−629055−3503−39][1001010−2001−5]
  15. Perform R1→R1+2R2: [−23−6−14351952−4−26][694374351952−4−26]
  16. Write in reduced row-echelon form: [2−6026−240−18−62−511][1001010−4001−5]
  17. Perform R2→R2+2R3: [4−11−1316−1333−1−11][4−11−13712−3−1933−1−11]
  18. Write in reduced row-echelon form: [−14421−4−5−6−46−3−6−6−45][100301020014]
  19. Perform R2→R2+3R1: [−44122−1−4][−4412−101132]
  20. Write in reduced row-echelon form: [35−610−1−10−4−14−51][100201020011]

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

R2→R2−3R1

  1. Take three times row 1.
  2. Subtract it from row 2, entry by entry.
  3. Rows 1 and 3 are untouched.

The answer is Only row 2 changes.

Where students go wrong

Changing the row you multiplied. In R2 → R2 − 3R1 it is row 2 that is replaced; row 1 is used and left exactly as it was. Overwriting row 1 as well loses the pivot you were about to need.

Also called augmented matrix, Gaussian elimination, row reduction or reduced row echelon form.

What you can put on this worksheet

Perform one row operation
Perform R1→R1+R2: [−211−224] → [−435−224]
Reduce to row-echelon form
Write in reduced row-echelon form: [1001−4−4] → [100011]
Solve a system by row reduction
Solve by row reduction: {−2x+y=1−2x+2y=4 → (1,3)

Questions about these worksheets

Yes. Every row operations sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the row operations questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: perform one row operation, reduce to row-echelon form and solve a system by row reduction. Tick as many as you want and set how many of each, or let it spread them evenly.

Changing the row you multiplied. In R2 → R2 − 3R1 it is row 2 that is replaced; row 1 is used and left exactly as it was. Overwriting row 1 as well loses the pivot you were about to need.