Cramer's Rule

Pronunciation: /ˈkreɪmərz rul/ ?

Cramer's rule is an algorithm for solving square linear systems using determinants.[1] 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

x+4y-2z=3, -2x+3y-3z=5, 2y-4z=8
Now convert this linear system into a 3x4 matrix.
3x4 matrix first row 1,4,-2,3; second row -2,3,-3,5; third row 0,2,-4,8
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.
determinant of 3x3 matrix named A0 first row 1,4,-2; second row -2,3,-3; third row 0,2,-4
Now find the value of |A0|.
determinant of A0 is 1*3*(-4)+4*(-3)*0+(-2)*(-2)*2-(-2)*3*0-4*(-2)*(-4)-1*(-3)*2=-12+0+8+0-32+6=-30
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|.
determinant of 3x3 matrix named Ax first row -3,4,-2; second row 5,3,-3; third row 8,2,-4
Now find the value of |Ax|.
determinant of Ax is (-3)*3*(-4)+4*(-3)*8+(-2)*5*2-(-2)*3*8-4*5*(-4)-(-3)*(-3)*2=36-96-20+48+80-18=30
Continue by finding the determinate of |Ay|. Do this by substituting the last column of matrix A into the second column of determinant |A0|.
determinant of 3x3 matrix named Ay first row 1,-3,-2; second row -2,5,-3; third row 0,8,-4
determinant of Ay is 1*5*(-4)+(-3)*(-3)*0+(-2)*(-2)*8-(-2)*5*0-(-3)*(-2)*(-4)-1*(-3)*8=-20+0+32+0+24+24=60
Now repeat the pattern for |Az|.
determinant of 3x3 matrix named Az first row 1,4,-3; second row -2,3,5; third row 0,2,8
determinant of Az is 1*3*8+4*5*0+(-3)*(-2)*2-(-3)*3*0-4*(-2)*8-1*5*2=24+0+12+0+64-10=90
The solution for the linear system is
x=|Ax|/|A0|, y=|Ay|/|A0|, z=|Az|/|A0|
So
x=30/(-30)=-1, y=60/-30=-2, z=90/-30=-3

References

  1. Cramer's rule. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2010-01-05). http://www.merriam-webster.com/dictionary/Cramer%27s rule.
  2. Skinner, Ernest Brown. College Algebra, pp 92-93. The Macmillan Company, 1917. (Accessed: 2010-01-22). http://www.archive.org/stream/cu31924031226503#page/n103/mode/1up/search/cramer.
  3. Householder, Alston S.. Principles Of Numerical Analysis, pp 28-29. McGraw-Hill Book Company, Inc., 1953. (Accessed: 2010-01-22). http://www.archive.org/stream/principlesofnume030218mbp#page/n47/mode/1up/search/cramer.

More Information

  • McAdams, David. Determinant. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Determinant%20of%20a%20Matrix.
  • McAdams, David. Matrix. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Matrix.
  • O'Connor, J J and Robertson, E F. Gabriel Cramer. Biographie. University of St Andrews, Scotland. 2009-03-12. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cramer.html.

Cite this article as:


Cramer's Rule. 2010-01-22. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/cramersrule.html.

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2010-01-22: Added "References" (McAdams, David.)
2008-12-27: Corrected url in citation (McAdams, David.)
2008-07-01: Initial version (McAdams, David.)

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