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4. Determinants of Square Matrices Cofactors Cramer's Rule Determinants and Cramer's Rule Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_46&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. The determinant of a 2-by-2 matrix [Equation 1] is defined to be [Equation 2]. Determinants of Square Matrices Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/determinants-of-square-matrices-206- 5864?campaign_content=book_196_section_46&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 1 View on Boundless.com Determinant as Area View on Boundless.com Equation 2 View on Boundless.com

6. The Laplace expansion for a n-by-n square matrix B is [Equation 3] for some fixed i or [Equation 4] for some fixed j. Determinants of Square Matrices Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/determinants-of-square-matrices-206- 5864?campaign_content=book_196_section_46&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 3 View on Boundless.com Determinant as Area View on Boundless.com Equation 4 View on Boundless.com

7. Recursively performing the Laplace expansion on each of the smaller sub- matrices in the Cofactor will eventually produce a small enough sub-matrix for which the determinant is known. Determinants of Square Matrices Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/determinants-of-square-matrices-206- 5864?campaign_content=book_196_section_46&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Determinant as Area View on Boundless.com Systems of Equations and Matrices > Determinants and Cramer's Rule

8. Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n. A k × k minor of A is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns. Cofactors Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cofactors-207- 5840?campaign_content=book_196_section_46&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Cofactor View on Boundless.com Systems of Equations and Matrices > Determinants and Cramer's Rule

9. The first minor of a matrix [Equation 5] is formed by removing the ith row and jth column of the matrix, and retrieving the determinant of the smaller matrix. Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cofactors-207- 5840?campaign_content=book_196_section_46&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 5 View on Boundless.com

10. The cofactor of an element [Equation 6] of a matrix A, written as [Equation 7] is defined as [Equation 8]. Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cofactors-207- 5840?campaign_content=book_196_section_46&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 6 View on Boundless.com Equation 7 View on Boundless.com Equation 8 View on Boundless.com

11. Cramer's Rule only works on square matrices that have a non-zero determinant and a unique solution. Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cramer-s-rule-208- 11096?campaign_content=book_196_section_46&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless Determinant as Area View on Boundless.com Systems of Equations and Matrices > Determinants and Cramer's Rule

12. Cramer's Rule is defined as [Equation 9], where [Equation 10] is the matrix formed by replacing the ith column of [Equation 11] by the column vector [Equation 12] in the equation. Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cramer-s-rule-208- 11096?campaign_content=book_196_section_46&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless Equation 9 View on Boundless.com Equation 10 View on Boundless.com Equation 11 View on Boundless.com

13. Cramer's Rule is efficient for solving small systems and can be calculated quite quickly; however, as the system grows, calculating the new determinants can be tedious. Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cramer-s-rule-208- 11096?campaign_content=book_196_section_46&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless Systems of Equations and Matrices > Determinants and Cramer's Rule

14. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

15. Key terms cofactor The cofactor of the (i, j) entry of a matrix is the signed minor of that entry. determinant The unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of 1 for the unit matrix. Its abbreviation is "det". minor A minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns. recursive of an expression, each term of which is determined by applying a formula to preceding terms square matrix A matrix having the same number of rows as columns. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

16. Determinant as Area The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Area parallellogram as determinant." Public domain http://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svg View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svgView on Boundless.com Systems of Equations and Matrices

17. Cofactor Here is a cofactor of an arbitrary 3x3 matrix taking the first row and second column out. Free to share, print, make copies and changes. Get yours at www.boundless.com Amazon Web Services. "Boundless." Public domain http://s3.amazonaws.com/figures.boundless.com/50bfa67ee4b0ce694712708f/20121205114403.jpg View on Boundless.comPublic domainhttp://s3.amazonaws.com/figures.boundless.com/50bfa67ee4b0ce694712708f/20121205114403.jpgView on Boundless.com Systems of Equations and Matrices

18. Determinant as Area The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Area parallellogram as determinant." Public domain http://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svg View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svgView on Boundless.com Systems of Equations and Matrices

19. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Use Cramer's Rule to solve for z in the following system: 2x+y+z=_______, x-y-z=0, x+2y+z=0. A) -2 B) 1 C) 0 D) 3

20. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices Use Cramer's Rule to solve for z in the following system: 2x+y+z=_______, x-y-z=0, x+2y+z=0. A) -2 B) 1 C) 0 D) 3

21. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Use Cramer's Rule to solve for y in the following system: 2x+y+z=3, x-y-z=0, x+2y+z=0. A) 3 B) 1 C) -2 D) 0

22. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices Use Cramer's Rule to solve for y in the following system: 2x+y+z=3, x-y-z=0, x+2y+z=0. A) 3 B) 1 C) -2 D) 0

23. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Use Cramer's Rule to solve for x in the following system: 2x+y+z=3, x-y-z=0, x+2y+z=0. A) 0 B) -2 C) 1 D) 3

24. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices Use Cramer's Rule to solve for x in the following system: 2x+y+z=3, x-y-z=0, x+2y+z=0. A) 0 B) -2 C) 1 D) 3

25. Attribution Wikipedia. "Minor (linear algebra)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Minor_(linear_algebra)CC BY-SA 3.0http://en.wikipedia.org/wiki/Minor_(linear_algebra) Wikipedia. "Cofactor (linear algebra)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Cofactor_(linear_algebra)CC BY-SA 3.0http://en.wikipedia.org/wiki/Cofactor_(linear_algebra) Wikipedia. "minor." CC BY-SA 3.0 http://en.wikipedia.org/wiki/minorCC BY-SA 3.0http://en.wikipedia.org/wiki/minor Wiktionary. "cofactor." CC BY-SA 3.0 http://en.wiktionary.org/wiki/cofactorCC BY-SA 3.0http://en.wiktionary.org/wiki/cofactor Wikipedia. "Determinant." CC BY-SA 3.0 http://en.wikipedia.org/wiki/DeterminantCC BY-SA 3.0http://en.wikipedia.org/wiki/Determinant Wikipedia. "Laplace expansion." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Laplace_expansionCC BY-SA 3.0http://en.wikipedia.org/wiki/Laplace_expansion Wiktionary. "recursive." CC BY-SA 3.0 http://en.wiktionary.org/wiki/recursiveCC BY-SA 3.0http://en.wiktionary.org/wiki/recursive Wiktionary. "cofactor." CC BY-SA 3.0 http://en.wiktionary.org/wiki/cofactorCC BY-SA 3.0http://en.wiktionary.org/wiki/cofactor Wiktionary. "determinant." CC BY-SA 3.0 http://en.wiktionary.org/wiki/determinantCC BY-SA 3.0http://en.wiktionary.org/wiki/determinant Wikipedia. "Determinant." CC BY-SA 3.0 http://en.wikipedia.org/wiki/DeterminantCC BY-SA 3.0http://en.wikipedia.org/wiki/Determinant Wikipedia. "Cramer's rule." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Cramer%2527s_ruleCC BY-SA 3.0http://en.wikipedia.org/wiki/Cramer%2527s_rule Wiktionary. "square matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/square+matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/square+matrix Wiktionary. "determinant." CC BY-SA 3.0 http://en.wiktionary.org/wiki/determinantCC BY-SA 3.0http://en.wiktionary.org/wiki/determinant Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices