# Cramer's Rule

Pronunciation: /ˈkreɪmərz rul/ Explain
Cramer's rule is an
algorithm
for solving square
linear systems
using
determinants of matrices.^{[2]}
Cramer's rule can only be used with linear systems that have
exactly
one
solution.
Cramer's rule is named after
Gabriel Cramer (1704–52),
a Swiss mathematician.

Cramer's rule uses determinants to find the solution of a linear system. For example,
start with the linear system

Now convert this linear system into a 3x4 matrix.

We will be making four determinants out of this matrix. Each of these determinants will
be based on a 3x3 square matrix. The first determinant, which we will label

`|A`_{0}|, is the first three columns of the matrix.

Now find the value of

`|A`_{0}|.

The second determinant will be based on the first. Substitute the last column
of matrix

`A` into the first column of determinant

`|A`_{0}|. The
new determinant is labeled

`|A`_{x}|.

Now find the value of

`|A`_{x}|.

Continue by finding the determinate of

`|A`_{y}|. Do this by
substituting the last column of matrix

`A` into the second column of
determinant

`|A`_{0}|.

Now repeat the pattern for

`|A`_{z}|.

The solution for the linear system is

So

### References

- Skinner, Ernest Brown. College Algebra. pp 92-93. www.archive.org. The Macmillan Company. 1917. Last Accessed 8/6/2018. http://www.archive.org/stream/cu31924031226503#page/n103/mode/1up/search/cramer. Buy the book
- Householder, Alston S.. Principles Of Numerical Analysis. pp 28-29. www.archive.org. McGraw-Hill Book Company, Inc.. 1953. Last Accessed 8/6/2018. http://www.archive.org/stream/principlesofnume030218mbp#page/n47/mode/1up/search/cramer. Buy the book

### More Information

- McAdams, David E.. Determinant. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/27/2018. http://www.allmathwords.org/en/d/determinant.html.

### Cite this article as:

McAdams, David E. Cramer's Rule. 6/25/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/cramersrule.html.
### Image Credits

### Revision History

6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (

McAdams, David E.)
1/22/2010: Added "References". (

McAdams, David E.)
12/27/2008: Corrected url in citation. (

McAdams, David E.)
7/1/2008: Initial version. (

McAdams, David E.)