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MATRICES AND SYSTEMS OF EQUATIONS

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1 MATRICES AND SYSTEMS OF EQUATIONS
Standards 2, 25 MATRICES AND SYSTEMS OF EQUATIONS INTRODUCTION ADDING MATRICES MULTIPLYING MATRICES INVERSE OF A MATRIX IDENTITY MATRIX SOLVING SYSTEMS WITH INVERSE MATRIX SOLVING EQUATIONS WITH AUGMENTED MATRICES END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

2 ALGEBRA II STANDARDS THIS LESSON AIMS:
Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices. Estándar 2: Los estudiantes resuelven sistemas de ecuaciones lineares y desigualdades (en 2 o tres variables) por substitución, con gráficas o con matrices. Standard 25: Students use properties from number systems to justify steps in combining and simplifying functions. Estándar 25: Los estudiantes usan propiedades de sistemas numéricos para justificar pasos en combinar y simplificar funciones. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3 This matrix B has dimensions 2X4
Standards 2, 25 MATRICES a 6 7 -2 5 y x columns rows B= This matrix B has dimensions 2X4 3 2 6 C= Matrix C is a column matrix of 3X1 x z D= Matrix D is a row matrix of 1X4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4 Multiplying by one scalar: Standards 2, 25
6 7 -2 2 10 12 14 -4 = Solve the following problems involving matrices: 2 5 3 -4 x 10 15 12 = 2x+1 6y-4 5 2 = 2x 5x 3x -4x 10 15 12 -4 = 2x+1=5 6y-4=2 +4 +4 -1 -1 2x = 10 5x = 15 3x = 12 -4x = -4 6y = 6 2x = 4 x= 5 x= 3 x= 4 x= 1 y= 1 x= 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5 Are matrices F and G equal?
Standards 2, 25 e 5 4 -3 E= e 5 4 -3 B= Matrix E and matrix B have the same dimensions 2X2 and the same elements, so they are equal. 3 4 7 -9 f z s -2 F= 2 1 G= Are matrices F and G equal? No, they have different number of columns and rows and different elements. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6 Standards 2, 25 ADDING MATRICES 2 3 -2 5 4 6 7 10 + = = 6 9 5 15 2+4
3+6 -2+7 5+10 6 8 7 2 4 10 - = = 4 -8 6-2 8-4 7-7 2-10 2 + = + = Observe that both matrices that are added or subtracted have the same dimensions. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7 Standards 2, 25 MULTIPLYING MATRICES
To multiply matrices the matrix at the left needs to have the same number of columns as rows have the one at the right, and the resulting matrix will have same number of rows as the one at the right and columns as the one at the left. 2 1 4 = (5)(2)+(7)(1)+(4)(4) = = 33 (1)(2)+(3)(1)+(2)(4) 13 It is possible 2X3 2X1 resulting matrix 3X1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8 Standards 2, 25 MULTIPLYING MATRICES 3 5 1 1 3 2 2 1 5 4 6 =
2 1 5 4 6 = (3)(2)+(5)(1)+(1)(5) (3)(4)+(5)(2)+(1)(6) (1)(2)+(3)(1)+(2)(5) (1)(4)+(3)(2)+(2)(6) It is possible = 2X3 2X2 resulting matrix 3X2 16 28 15 22 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9 Calculate the A for matrix A:
Standards 2, 25 Calculate the A for matrix A: -1 8 2 6 4 A= 8 2 6 4 = (8)(4) –(6)(2) =32 -12 =20 = 4 20 -2 -6 8 = 1 5 -1 10 -3 2 4 -2 -6 8 1 20 A = -1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10 Calculate the A for matrix A:
Standards 2, 25 Calculate the A for matrix A: -1 4 3 5 1 A= 4 3 5 1 = (4)(1) –(5)(3) =4 -15 = -11 = -1 11 3 5 -4 1 -3 -5 4 1 -11 A = -1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11 Verify the identity property above indicated for matrix A below:
Standards 2, 25 IDENTITY MATRIX A I = I A = A Verify the identity property above indicated for matrix A below: 8 2 6 4 A= 1 I= Diagonal 1 8 2 6 4 I A = = 1(8) + 0(6) 1(2) + 0(4) 0(8) + 1(6) 0(2) + 1(4) 8 2 6 4 = 1 8 2 6 4 A I = = 8(1) + 6(0) 2(1) + 4(0) 8(0) + 6(1) 2(0) + 4(1) 8 2 6 4 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12 Write the system of equations represented by each matrix equation:
Standards 2, 25 Write the system of equations represented by each matrix equation: -3 6 7 1 x y = 15 -8 -3x + 6y = 15 7x + y = -8 5 9 -2 4 x y = 5x + 9y = 0 -2x + 4y = 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

13 Multiplying both sides by the inverse:
Standards 2, 25 Solve the following system of equations using matrices: 4x + 2y = 10 5x + y = 17 1 Multiplying both sides by the inverse: Write as matrix equation: -1 6 1 3 5 -2 = -1 6 1 3 5 -2 4 2 5 1 x y = 10 17 4 2 5 1 x y 10 17 Finding the determinant of the coefficient matrix: 4 2 5 1 = (4)(1) –(5)(2) =4 -10 = -6 Finding the inverse of the coefficient matrix: = -1 6 2 5 -4 = -1 6 1 3 5 -2 1 -2 -5 4 1 -6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14 Solve the following system of equations using matrices:
Standards 2, 25 Solve the following system of equations using matrices: 4x + 2y = 10 5x + y = 17 1 Write as matrix equation: -1 6 1 3 5 -2 -1 6 1 3 5 -2 4 2 5 1 x y 10 17 4 2 5 1 x y 10 17 = = -1 6 1 3 (4) (5) + -1 6 1 3 (2) (1) + -1 6 1 3 (10) (17) + x y = 5 6 -2 3 (10) (17) + 5 6 -2 3 (4) (5) + 5 6 -2 3 (2) (1) + 1 x y = 4 -3 x y = 4 -3 Solution is (4,-3) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15 Multiplying both sides by the inverse:
Standards 2, 25 Solve the following system of equations using matrices: 2x + 5y = 13 6x + 3y = 3 Multiplying both sides by the inverse: Write as matrix equation: -1 8 5 24 1 4 12 = -1 8 5 24 1 4 12 2 5 6 3 x y = 13 3 2 5 6 3 x y 13 3 Finding the determinant of the coefficient matrix: 2 5 6 3 = (2)(3) –(6)(5) =6 -30 = -24 Finding the inverse of the coefficient matrix: = -3 24 5 6 -2 = -1 8 5 24 1 4 12 3 -5 -6 2 1 -24 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16 Solve the following system of equations using matrices:
Standards 2, 25 Solve the following system of equations using matrices: 2x + 5y = 13 6x + 3y = 3 Write as matrix equation: -1 8 5 24 1 4 12 -1 8 5 24 1 4 12 2 5 6 3 x y 13 3 2 5 6 3 x y 13 3 = = -1 8 5 24 (2) (6) + -1 8 5 24 (5) (3) + -1 8 5 24 (13) ( 3) + = x y 1 4 -1 12 (13) ( 3) + 1 4 -1 12 (2) (6) + 1 4 -1 12 (5) (3) + 1 x y = -1 3 x y = -1 3 Solution is (-1,3) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

17 Write the augmented matrix for this system, then reduce it to solve it:
Standards 2, 25 1 3 x 3x- 2y + z = 2 2x+3y -4z = -4 . (2) 4x+ 2y -2z = 2 -2 3 1 3 2 3 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18 Write the augmented matrix for this system, then reduce it to solve it:
Standards 2, 25 3x- 2y + z = 2 2x+3y -4z = -4 + 4x+ 2y -2z = 2 -2 3 1 3 2 3 -2(1) +2 = 0 x -2 1 -2( ) + 3 = -2 3 13 13 3 -14 3 -16 3 = -2( ) - 4 = 1 3 -14 -2( ) - 4 = -16 3 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

19 Write the augmented matrix for this system, then reduce it to solve it:
Standards 2, 25 3x- 2y + z = 2 2x+3y -4z = -4 4x+ 2y -2z = 2 + -2 3 1 3 2 3 x -2 1 13 3 -14 3 -16 3 = 7 3 -5 3 -1 3 -2(1) +2 = 0 -2( ) + 1 = -2 3 7 -2( ) -1 = 1 3 - 5 -2( ) +1 = - 1 3 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

20 Write the augmented matrix for this system, then reduce it to solve it:
Standards 2, 25 3x- 2y + z = 2 2x+3y -4z = -4 4x+ 2y -2z = 2 -2 3 1 3 2 3 1 3 13 x 13 3 -14 3 -16 3 7 3 -5 3 -1 3 -14 13 -16 13 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

21 Write the augmented matrix for this system, then reduce it to solve it:
Standards 2, 25 3x- 2y + z = 2 2x+3y -4z = -4 4x+ 2y -2z = 2 -2 3 1 3 2 3 1 + 13 3 -14 3 -16 3 7 3 -5 3 -1 3 -15 39 -6 39 1 = 2 3 x -14 13 -16 13 1 2 3 -14 13 1 + = -15 39 2 3 -16 13 + = - 6 39 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

22 Write the augmented matrix for this system, then reduce it to solve it:
Standards 2, 25 3x- 2y + z = 2 2x+3y -4z = -4 4x+ 2y -2z = 2 -2 3 1 3 2 3 1 - 7 3 -14 13 5 - = 11 13 3 -14 3 -16 3 7 3 -5 3 -1 3 - 7 3 -16 13 1 - = 99 39 + -15 39 -6 39 1 -14 13 -7 3 x -16 13 1 11 13 99 39 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

23 Write the augmented matrix for this system, then reduce it to solve it:
Standards 2, 25 3x- 2y + z = 2 2x+3y -4z = -4 4x+ 2y -2z = 2 -2 3 1 3 2 3 1 13 3 -14 3 -16 3 7 3 -5 3 -1 3 -15 39 -6 39 1 -14 13 -16 13 1 13 11 x 11 13 99 39 1 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

24 Write the augmented matrix for this system, then reduce it to solve it:
Standards 2, 25 3x- 2y + z = 2 2x+3y -4z = -4 4x+ 2y -2z = 2 -2 3 1 3 2 3 1 13 3 -14 3 -16 3 7 3 -5 3 -1 3 -15 39 -6 39 1 14 13 16 - = 3 2 -14 13 -16 13 1 + 11 13 99 39 1 2 = 14 13 x 1 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

25 Write the augmented matrix for this system, then reduce it to solve it:
Standards 2, 25 3x- 2y + z = 2 2x+3y -4z = -4 4x+ 2y -2z = 2 -2 3 1 3 2 3 1 13 3 -14 3 -16 3 7 3 -5 3 -1 3 -15 39 -6 39 1 + -14 13 -16 13 1 15 39 6 - = 3 1 11 13 99 39 1 1 = 1 2 15 39 x 1 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

26 Write the augmented matrix for this system, then reduce it to solve it:
Standards 2, 25 1 3 x 3x- 2y + z = 2 2x+3y -4z = -4 . (2) 4x+ 2y -2z = 2 + + -2 3 1 3 2 3 -2 +2 = 0 x -2 1 -2( ) + 3 = -2 3 13 - 7 3 -14 13 5 - = 11 3 13 x 13 3 -14 3 -16 3 = = -2( ) - 4 = 1 3 -14 7 3 -5 3 -1 3 + -2( ) - 4 = -16 3 2 - 7 3 -16 13 1 - = 99 39 -15 39 -6 39 + 1 = -2 +2 = 0 14 13 16 - = 3 2 2 3 x -14 13 -7 3 x -16 13 -2( ) + 1 = -2 3 7 1 -2( ) -1 = 1 3 - 5 15 39 6 - = 3 1 13 11 x 11 13 99 39 = + -2( ) +1 = - 1 3 2 + 1 1 = 1 2 2 3 -14 13 1 + = -15 39 = 14 13 x 15 39 x 1 3 2 3 -16 13 + = - 6 39 The solution is (1,2,3) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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