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

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

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 5 -2 x z D= Matrix D is a row matrix of 1X4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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 5 5 3 3 -4 -4 6 6 2 2 2 2 x= 5 x= 3 x= 4 x= 1 y= 1 x= 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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

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 3 -2 4 1 2 1 0 2 3 + = 6 -4 8 2 1 0 2 3 + 7 -4 10 5 = Observe that both matrices that are added or subtracted have the same dimensions. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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. 5 7 4 1 3 2 2 1 4 = (5)(2)+(7)(1)+(4)(4) = 10 + 7 + 16 = 33 (1)(2)+(3)(1)+(2)(4) 2 + 3 + 8 13 It is possible 2X3 2X1 resulting matrix 3X1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standards 2, 25 MULTIPLYING MATRICES 3 5 1 1 3 2 2 1 5 4 6 = 3 5 1 1 3 2 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 = 6 + 5 + 5 12 + 10 + 6 2X3 2X2 resulting matrix 3X2 2 + 3 + 10 4 + 6 + 12 16 28 15 22 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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

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

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

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

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

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

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

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

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

Write the augmented matrix for this system, then reduce it to solve it: Standards 2, 25 3x- 2y + z = 2 3 -2 1 2 3 -2 1 2 2x+3y -4z = -4 2 3 -4 -4 2 3 -4 -4 + 4x+ 2y -2z = 2 4 2 -2 2 2 1 -1 1 -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

Write the augmented matrix for this system, then reduce it to solve it: Standards 2, 25 3x- 2y + z = 2 3 -2 1 2 3 -2 1 2 2x+3y -4z = -4 2 3 -4 -4 2 3 -4 -4 4x+ 2y -2z = 2 4 2 -2 2 2 1 -1 1 + -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

Write the augmented matrix for this system, then reduce it to solve it: Standards 2, 25 3x- 2y + z = 2 3 -2 1 2 3 -2 1 2 2x+3y -4z = -4 2 3 -4 -4 2 3 -4 -4 4x+ 2y -2z = 2 4 2 -2 2 2 1 -1 1 -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

Write the augmented matrix for this system, then reduce it to solve it: Standards 2, 25 3x- 2y + z = 2 3 -2 1 2 3 -2 1 2 2x+3y -4z = -4 2 3 -4 -4 2 3 -4 -4 4x+ 2y -2z = 2 4 2 -2 2 2 1 -1 1 -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

Write the augmented matrix for this system, then reduce it to solve it: Standards 2, 25 3x- 2y + z = 2 3 -2 1 2 3 -2 1 2 2x+3y -4z = -4 2 3 -4 -4 2 3 -4 -4 4x+ 2y -2z = 2 4 2 -2 2 2 1 -1 1 -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

Write the augmented matrix for this system, then reduce it to solve it: Standards 2, 25 3x- 2y + z = 2 3 -2 1 2 3 -2 1 2 2x+3y -4z = -4 2 3 -4 -4 2 3 -4 -4 4x+ 2y -2z = 2 4 2 -2 2 2 1 -1 1 -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

Write the augmented matrix for this system, then reduce it to solve it: Standards 2, 25 3x- 2y + z = 2 3 -2 1 2 3 -2 1 2 2x+3y -4z = -4 2 3 -4 -4 2 3 -4 -4 4x+ 2y -2z = 2 4 2 -2 2 2 1 -1 1 -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

Write the augmented matrix for this system, then reduce it to solve it: Standards 2, 25 3x- 2y + z = 2 3 -2 1 2 3 -2 1 2 2x+3y -4z = -4 2 3 -4 -4 2 3 -4 -4 4x+ 2y -2z = 2 4 2 -2 2 2 1 -1 1 -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

Write the augmented matrix for this system, then reduce it to solve it: Standards 2, 25 1 3 x 3x- 2y + z = 2 3 -2 1 2 3 -2 1 2 2x+3y -4z = -4 2 3 -4 -4 2 3 -4 -4 . (2) 4x+ 2y -2z = 2 + 4 2 -2 2 2 1 -1 1 + -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