Skip to main content
Mathovia

Math Practice

Augmented Matrix Practice Problems

Write each system as an augmented matrix, use row operations to reach row-echelon or reduced row-echelon form, and read off the solution. A few problems ask you to interpret rows like 0 0 | 7.

  1. 1.

    Write the augmented matrix of and . Type the rows separated by a semicolon, like "2, -1 | 7; 1, 3 | -4".

    Show full solution

    One row per equation, columns , , then the constant after the bar: and .

  2. 2.

    The augmented matrix has rows and . What is ?

    Show full solution

    The second row is , so .

  3. 3.

    A matrix in reduced row-echelon form has rows and . What is the solution?

    Show full solution

    Each row reads directly: and .

  4. 4.

    A system’s matrix reduces to rows and . How many solutions does it have? Answer 0, 1 or "infinitely many".

    Show full solution

    The second row says , which is false, so the system is inconsistent.

  5. 5.

    A system’s matrix reduces to rows and . How many solutions does it have? Answer 0, 1 or "infinitely many".

    Show full solution

    A row of zeros says . Only remains, so is free and there are infinitely many solutions.

  6. 6.

    In the matrix with rows and , what does the second row become after ? Type it like "0, -7 | -14".

    Show full solution

    , , .

  7. 7.

    Solve and with an augmented matrix.

    Show full solution

    : . Divide by −7: . : . So .

  8. 8.

    Solve and with an augmented matrix.

    Show full solution

    : , so . Then (after dividing by 3) gives .

  9. 9.

    Solve and with an augmented matrix.

    Show full solution

    : . : , so ; then .

  10. 10.

    Solve , , by Gaussian elimination.

    Show full solution

    and , swap, then gives : , then , .

  11. 11.

    Which system does the matrix with rows ; ; represent? What is the first equation?

    Show full solution

    Attach to the columns and drop the zero terms: , , .

  12. 12.

    For the matrix with rows and , the general solution is , . What is when ?

    Show full solution

    , giving the solution . Check: ✓.
Written by
MPhil in Mathematics, Founder of Mathovia
Published

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Augmented Matrix Practice Problems

These problems follow the augmented matrices lesson. Write out every intermediate matrix: most mistakes come from changing only part of a row.

Row operations

OperationNotation
swap two rows
multiply a row by a non-zero number
add a multiple of one row to another

Common mistakes to watch for

  • Misaligned columns. Write every equation in the same variable order, with 0 for any missing variable.
  • Forgetting the constants column. The augmented column holds the right-hand sides and changes with every row operation.
  • Illegal row operations. Allowed moves: swap rows, multiply a row by a nonzero number, add a multiple of one row to another.
  • Misreading the final matrix. A row means no solution; a row of zeros means a free variable.

Frequently Asked Questions

How do I write an augmented matrix?+

One row per equation, one column per variable in the same order, and the constants after the bar. Use 0 for a missing variable.

What does a row 0 0 | c mean?+

If c is not zero, the system has no solution. If c is zero, that equation added no information, which usually means infinitely many solutions.

How do I type a matrix row?+

Separate entries with commas and rows with a semicolon, for example 2, -1 | 7; 1, 3 | -4.

Calculators for this topic

Gaussian Elimination Calculator

More practice problems

Systems With Three Variables Practice Problems Elimination Method Practice Problems Systems of Equations Practice Problems