What is Cramer's rule?

A determinant-based formula for finding each unknown in a square linear system that has a unique solution.

What does $A$ contain in $A\mathbf{x}=\mathbf{b}$?

$A$ is the square coefficient matrix; its entries are the coefficients of the unknowns.

It is the matrix obtained from $A$ by replacing its $i$th column with the right-hand-side vector $\mathbf{b}$.

What is the formula for $x_i$?

$x_i=\det(A_i)/\det(A)$, provided $\det(A)\ne 0$.

Why is a nonzero coefficient determinant important?

It makes the coefficient matrix invertible and ensures the square system has a unique solution, so the determinant ratios are defined.

Linear algebraLesson 1 of 1 in the course Systems of linear equations

Cramer's Rule: Determinant Formulas, Derivation, and Limitations

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

Cramer's rule gives an explicit determinant-based formula for solving a square system of linear equations when that system has exactly one solution. This lesson introduces the matrix setup, the rule's key condition, and how to construct the determinants used in the formula.

A system with $n$ equations and $n$ unknowns can be written as $A\mathbf{x}=\mathbf{b}$ . Here, $A$ is the $n\times n$ coefficient matrix, $\mathbf{x}=(x_1,\ldots,x_n)^T$ is the column of unknowns, and $\mathbf{b}$ is the column of right-hand-side constants.

All flashcards

What is Cramer's rule?: A determinant-based formula for finding each unknown in a square linear system that has a unique solution.
What does $A$ contain in $A\mathbf{x}=\mathbf{b}$ ?: $A$ is the square coefficient matrix; its entries are the coefficients of the unknowns.
What is $A_i$ ?: It is the matrix obtained from $A$ by replacing its $i$ th column with the right-hand-side vector $\mathbf{b}$ .
What is the formula for $x_i$ ?: $x_i=\det(A_i)/\det(A)$ , provided $\det(A)\ne 0$ .
Why is a nonzero coefficient determinant important?: It makes the coefficient matrix invertible and ensures the square system has a unique solution, so the determinant ratios are defined.

Quiz questions

What shape must the coefficient matrix have for standard Cramer's rule?Answer: A square matrix, with equal numbers of equations and unknowns.
What condition on the coefficient determinant allows Cramer's rule to give the unique solution?Answer: $\det(A)\ne 0$ .
If $\det(A)=0$ , what can be concluded about a square linear system?Answer: It has no unique solution; it may be inconsistent or have infinitely many solutions.
For an invertible matrix $A$ , which identity expresses its inverse using the adjugate?Answer: $A^{-1}=\operatorname{adj}(A)/\det(A)$ .
What does substituting a proposed Cramer's rule solution into the original equations check?Answer: Whether the proposed values satisfy every original equation.

Based on the Wikipedia article Cramer's rule, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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