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Math Practice

Cramer's Rule Practice Problems

Find the determinant of the coefficients, swap the constants into one column at a time, and divide. The early problems practise the determinants on their own.

  1. 1.

    Evaluate the determinant with top row 3, 2 and bottom row 1, 4.

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    .

  2. 2.

    Evaluate the determinant with top row 5, −2 and bottom row 3, 1.

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    .

  3. 3.

    Evaluate the determinant with top row −4, 6 and bottom row 2, −3.

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    .

  4. 4.

    For and , find the coefficient determinant .

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  5. 5.

    Use Cramer’s rule to solve and .

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    , , , so and .

  6. 6.

    Use Cramer’s rule to solve and .

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    , , , so and .

  7. 7.

    Use Cramer’s rule to solve and .

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    , , , so and .

  8. 8.

    For and , find , the determinant with the -column replaced by the constants.

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    has top row 1, 2 and bottom row 5, −1: .

  9. 9.

    Use Cramer’s rule to solve and .

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    , , , so and .

  10. 10.

    Does and have one solution, no solution, or infinitely many?

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    but , so the lines are parallel: no solution.

  11. 11.

    Evaluate the 3 × 3 determinant with rows 1, 0, 2 / 3, 1, 0 / 0, 2, 1.

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    Expand along the top row: .

  12. 12.

    For , , , the coefficient determinant is and . Find .

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    .
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MPhil in Mathematics, Founder of Mathovia
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Cramer’s Rule Practice Problems

These problems follow the Cramer’s rule lesson. Write each equation in standard form before reading off the coefficients.

The rule

Common mistakes to watch for

  • Replacing the wrong column. For , replace the -column of the coefficient matrix with the constants; for , the -column.
  • Sign errors in determinants. For a 2×2, . For a 3×3, follow the sign pattern on the first row.
  • Dividing by a zero determinant. If , Cramer’s rule cannot be used; the system has no solution or infinitely many.
  • Not writing equations in standard form first. Line up , and constants before reading off the coefficients.

Frequently Asked Questions

How do I type my answers?+

Type determinants as numbers and solutions as ordered pairs like (2, 3). For the classification question, type one solution, no solution or infinitely many.

What do I do if D is zero?+

Cramer's rule doesn't apply. Check the other determinants: if any is non-zero there is no solution; if they are all zero there are usually infinitely many.

More practice problems

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