cramer's rule: meaning, definition, pronunciation and examples
Rare / TechnicalTechnical / Academic / Mathematical
Quick answer
What does “cramer's rule” mean?
A mathematical theorem in linear algebra that gives an explicit formula for the solution of a system of linear equations with as many equations as unknowns, provided the system's coefficient matrix has a non-zero determinant.
Audio
Pronunciation
Definition
Meaning and Definition
A mathematical theorem in linear algebra that gives an explicit formula for the solution of a system of linear equations with as many equations as unknowns, provided the system's coefficient matrix has a non-zero determinant.
A method for solving a system of simultaneous linear equations using determinants of matrices. It provides a theoretical formula for each unknown variable as a ratio of two determinants: the determinant of a modified coefficient matrix divided by the determinant of the original coefficient matrix.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or grammatical differences exist for this technical term. Both dialects use the possessive apostrophe-s. Spelling of related terms like 'formulae' vs. 'formulas' may vary by writer preference.
Connotations
Identical in both dialects. Connotes precision, theoretical mathematics, and linear algebra. It is computationally inefficient for large systems, so it is often taught for its theoretical insight rather than practical application.
Frequency
Equally rare and specialized in both dialects. Its use is confined almost entirely to textbooks, courses, and papers on linear algebra.
Grammar
How to Use “cramer's rule” in a Sentence
Cramer's rule + verb (states, provides, gives)verb (use, apply, derive) + Cramer's ruleSolution/unknown + via + Cramer's ruleVocabulary
Collocations
Examples
Examples of “cramer's rule” in a Sentence
verb
British English
- The system can be cramered to find the values of x and y.
- After setting up the matrices, we proceeded to cramer the unknowns.
American English
- We Cramer'd the system as an exercise.
- The software library includes a function to Cramer the solution.
Usage
Meaning in Context
Business
Virtually never used.
Academic
Used exclusively in mathematics, physics, and engineering contexts when discussing theoretical solutions to linear systems. Common in linear algebra textbooks and courses.
Everyday
Not used.
Technical
The primary context. Refers to a specific algorithm. Usage often accompanied by caveats about its computational complexity.
Vocabulary
Synonyms of “cramer's rule”
Neutral
Weak
Vocabulary
Antonyms of “cramer's rule”
Watch out
Common Mistakes When Using “cramer's rule”
- Misspelling as 'Cramer rule' (omitting apostrophe-s).
- Pronouncing 'Cramer' with a short /æ/ as in 'cram' instead of the long /eɪ/.
- Attempting to apply it to systems where the number of equations does not equal the number of unknowns, or where the determinant is zero.
- Using it as a general synonym for 'method' or 'procedure'.
FAQ
Frequently Asked Questions
It is named after the Swiss mathematician Gabriel Cramer (1704–1752), who published the rule in his 1750 work 'Introduction à l'analyse des lignes courbes algébriques'.
It is primarily used for small systems (2x2 or 3x3) for its instructive clarity or in theoretical proofs. For larger systems or practical computation, algorithms like Gaussian elimination or matrix decomposition are vastly more efficient.
The system must be square (same number of equations as unknowns) and have a non-zero determinant for its coefficient matrix. If the determinant is zero, the system either has no solution or infinitely many, and Cramer's rule cannot be applied.
Almost never for serious numerical computation due to its poor scaling (O(n!) complexity). However, it might be implemented for educational purposes or for solving very small, symbolic systems in computer algebra systems.
A mathematical theorem in linear algebra that gives an explicit formula for the solution of a system of linear equations with as many equations as unknowns, provided the system's coefficient matrix has a non-zero determinant.
Cramer's rule is usually technical / academic / mathematical in register.
Cramer's rule: in British English it is pronounced /ˈkreɪ.məz ˌruːl/, and in American English it is pronounced /ˈkreɪ.mɚz ˌrul/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
CRAM-er's Rule: To find the answer, you CRAM the column of constants into the matrix, then compute the RATIO of determinants.
Conceptual Metaphor
A PRECISE RECIPE: The rule is a fixed, step-by-step formula that guarantees a specific result, much like a chemical formula or a cooking recipe for solving equations.
Practice
Quiz
What is a major practical limitation of Cramer's rule?