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4. The Identity Matrix The Inverse of a Matrix Solving Systems of Equations Using Matrices Inverses of Matrices Systems of Equations and Matrices > Inverses of Matrices Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_45&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. For any square matrix, its identity matrix is a diagonal stretch of 1s going from the upper-left-hand corner to the lower-right, with all other elements being 0. Non-square matrices do not have an identity. That is, for a non-square matrix [A], there is no matrix such that [A][I]=[I][A]=[A]. Proving that the identity matrix functions as desired requires the use of matrix multiplication. The Identity Matrix 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/inverses-of-matrices-45/the-identity- matrix-203- 5883?campaign_content=book_196_section_45&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Matrix multiplication View on Boundless.com Systems of Equations and Matrices > Inverses of Matrices

6. Note that, just as in the definition of the identity matrix, this definition requires commutativity—the multiplication must work the same in either order. To be invertible, a matrix must be square, because the identity matrix must be square as well. The Inverse of a Matrix 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/inverses-of-matrices-45/the-inverse-of- a-matrix-204- 8065?campaign_content=book_196_section_45&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Multiplying matrices View on Boundless.com Systems of Equations and Matrices > Inverses of Matrices

7. To determine the inverse of the matrix [Equation 1], set [Equation 2]. Then solve for a, b, c, and d. Systems of Equations and Matrices > Inverses of 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/inverses-of-matrices-45/the-inverse-of- a-matrix-204- 8065?campaign_content=book_196_section_45&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 1 View on Boundless.com Equation 2 View on Boundless.com

8. Using matrices to solve systems of equations can drastically reduce the workload on you. Consider the following three equations: [Equation 3][Equation 4][Equation 5]. Solving Systems of Equations Using Matrices Systems of Equations and Matrices > Inverses of 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/inverses-of-matrices-45/solving- systems-of-equations-using-matrices-205- 5539?campaign_content=book_196_section_45&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 3 View on Boundless.com Matrix multiplication View on Boundless.com Equation 4 View on Boundless.com Equation 5 View on Boundless.com

9. To solve these equations using matrices, we first define a 3×3 matrix [A], which is the coefficients of all the variables on the left side of the equal signs: [Equation 6]Also, define a 3×1 matrix [B], which is the numbers on the right side of the equal signs: [Equation 7]. Solving Systems of Equations Using Matrices Systems of Equations and Matrices > Inverses of 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/inverses-of-matrices-45/solving- systems-of-equations-using-matrices-205- 5539?campaign_content=book_196_section_45&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 6 View on Boundless.com Matrix multiplication View on Boundless.com Equation 7 View on Boundless.com

10. In order to determine the values of x, y, and z, we simply multiply the inverse of [A] times [B]. This is most readily done using a calculator. The calculator responds with a 3×1 matrix, which is all three answers. In this case, x=3, y=5, and z=2. Solving Systems of Equations Using 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/inverses-of-matrices-45/solving- systems-of-equations-using-matrices-205- 5539?campaign_content=book_196_section_45&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Matrix multiplication View on Boundless.com Systems of Equations and Matrices > Inverses of Matrices

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

12. Key terms identity matrix A diagonal matrix all of the diagonal elements of which are equal to 1, the rest being equal to 0. inverse matrix Of a matrix [A], another matrix [B] such that [A] multiplied by [B] and [B] multiplied by [A] both equal the identity matrix. linear equation A polynomial equation of the first degree (such as x = 2y - 7). matrix A rectangular arrangement of numbers or terms having various uses such as transforming coordinates in geometry, solving systems of linear equations in linear algebra and representing graphs in graph theory. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

13. Multiplying matrices Remember, when multiplying matrices, that one mnemonic to remember which terms to multiply together is to pretend that you have placed the row from the first matrix onto a dump truck, and are "dumping" the terms onto the column from the second matrix. Free to share, print, make copies and changes. Get yours at www.boundless.com Connexions. "Matrix Concepts -- Multiplying Matrices." CC BY 3.0 http://cnx.org/content/m18291/latest/?collection=col11354/latest View on Boundless.comCC BY 3.0http://cnx.org/content/m18291/latest/?collection=col11354/latestView on Boundless.com Systems of Equations and Matrices

14. Matrix multiplication When doing matrix multiplication, it may help to remember the mnemonic device of taking the rows of the first matrix and "dumping" them into the columns of the second matrix. Free to share, print, make copies and changes. Get yours at www.boundless.com Connexions. "Multiplying Matrices." CC BY 3.0 http://cnx.org/content/m18291/latest/?collection=col10624 View on Boundless.comCC BY 3.0http://cnx.org/content/m18291/latest/?collection=col10624View on Boundless.com Systems of Equations and Matrices

15. Matrix multiplication When doing matrix multiplication, it may help to remember the mnemonic device of taking the rows of the first matrix and "dumping" them into the columns of the second matrix. Free to share, print, make copies and changes. Get yours at www.boundless.com Connexions. "Multiplying Matrices." CC BY 3.0 http://cnx.org/content/m18291/latest/?collection=col10624 View on Boundless.comCC BY 3.0http://cnx.org/content/m18291/latest/?collection=col10624View on Boundless.com Systems of Equations and Matrices

16. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Use inverse matrices to solve the following system of linear equations: x+3y=6, x-2z=-1, and 3y+z=5. A) x=2, y=1, z=3 B) x=4, y=5, z=2 C) x=3, y=1, z=2 D) x=6, y=1, z=-3

17. 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 inverse matrices to solve the following system of linear equations: x+3y=6, x-2z=-1, and 3y+z=5. A) x=2, y=1, z=3 B) x=4, y=5, z=2 C) x=3, y=1, z=2 D) x=6, y=1, z=-3

18. Attribution Connexions. "Solving Linear Equations." CC BY 3.0 http://cnx.org/content/m18292/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m18292/latest/?collection=col10624 Wiktionary. "linear equation." CC BY-SA 3.0 http://en.wiktionary.org/wiki/linear+equationCC BY-SA 3.0http://en.wiktionary.org/wiki/linear+equation Wiktionary. "inverse matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/inverse+matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/inverse+matrix Wikipedia. "Identity matrix." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Identity_matrixCC BY-SA 3.0http://en.wikipedia.org/wiki/Identity_matrix Connexions. "Multiplying Matrices." CC BY 3.0 http://cnx.org/content/m18291/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m18291/latest/?collection=col10624 Connexions. "The Identity Matrix." CC BY 3.0 http://cnx.org/content/m18293/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m18293/latest/?collection=col10624 Wiktionary. "identity matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/identity+matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/identity+matrix Wiktionary. "matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/matrix Wikipedia. "Inverse matrix." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Inverse_matrixCC BY-SA 3.0http://en.wikipedia.org/wiki/Inverse_matrix Connexions. "The Inverse Matrix." CC BY 3.0 http://cnx.org/content/m18294/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m18294/latest/?collection=col10624 Wiktionary. "identity matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/identity+matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/identity+matrix Wiktionary. "inverse matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/inverse+matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/inverse+matrix Wiktionary. "matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/matrix Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices