Matrices
Cramer's rule 2 × 2 worksheets
Solve twelve two-variable systems with Cramer's rule. Practice unique, infinitely many and no-solution systems with exact determinant reasoning.
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Cramer's rule 3 × 3 worksheets
Solve twelve three-variable systems with Cramer's rule. Use exact determinants and rank evidence for unique, infinitely many and no-solution systems.
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Write D, Dx, Dy and Dz. If D is nonzero, calculate x = Dx/D, y = Dy/D and z = Dz/D. If D is zero, row-reduce the augmented system and compare ranks to determine whether there are infinitely many solutions or no solution.
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Customize this worksheetLet the main determinant choose the path
Build D from the coefficient matrix, then replace one coefficient column at a time with the constants to form Dx, Dy and Dz. When D ≠ 0, each variable is its replacement determinant divided by D, kept as an exact integer or fraction.
When D = 0, Cramer's quotients are undefined and the system needs a consistency check. Row-reduce the augmented matrix: equal ranks below the number of variables mean infinitely many solutions; a larger augmented rank means no solution.
Even D = Dx = Dy = Dz = 0 does not, by itself, prove infinitely many solutions. The reduced augmented matrix supplies the missing rank evidence.
Learning objective
What this activity builds
Use four 3 × 3 determinants for unique solutions and exact augmented reduction to classify singular three-variable systems.
Worked example
Solve x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2.
x = 1, y = 2 and z = 3.
D = 7, Dx = 7, Dy = 14 and Dz = 21. Therefore x = Dx/D = 1, y = Dy/D = 2 and z = Dz/D = 3.
Teaching notes
Make the practice count
- Align x, y, z and the constants before copying entries into matrices.
- Label D, Dx, Dy and Dz so each replacement column stays clear.
- Check a unique solution by substituting it into all three original equations.
- For D = 0, use the reduced augmented matrix and ranks as the classification evidence.
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