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 |A0|, is the first three columns of the matrix.
Now find the value of |A0|.
The second determinant will be based on the first. Substitute the last column of matrix A into the first column of determinant |A0|. The new determinant is labeled |Ax|.
Now find the value of |Ax|.
Continue by finding the determinate of |Ay|. Do this by substituting the last column of matrix A into the second column of determinant |A0|.
Now repeat the pattern for |Az|.
The solution for the linear system is
So

References

1. McAdams, David E.. All Math Words Dictionary, Cramer's rule. 2nd Classroom edition 20150108-4799968. pg 50. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Skinner, Ernest Brown. College Algebra. pp 92-93. www.archive.org. The Macmillan Company. 1917. Last Accessed 6/25/2018. http://www.archive.org/stream/cu31924031226503#page/n103/mode/1up/search/cramer. Buy the book
3. Householder, Alston S.. Principles Of Numerical Analysis. pp 28-29. www.archive.org. McGraw-Hill Book Company, Inc.. 1953. Last Accessed 6/25/2018. http://www.archive.org/stream/principlesofnume030218mbp#page/n47/mode/1up/search/cramer. Buy the book

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

McAdams, David E. Cramer's Rule. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/c/cramersrule.html.

Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)