Matrices: Multiplication, Determinants, Inversion, & More
Explore the world of matrices with topics like matrix multiplication, determinants, inverting matrices, and solving systems of equations. Learn about matrix sizes, identity matrices, and how to find products. Dive into the realm of mathematics with practical examples and exercises.
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6) Matrices 6.1) Introduction to matrices 6.2) Matrix multiplication 6.3) Determinants 6.4) Inverting a 2 x 2 matrix 6.5) Inverting a 3 x 3 matrix 6.6) Solving systems of equations using matrices
Chapter CONTENTS 6.1) Introduction to matrices
Your turn Worked example Write down the size of the matrix: Write down the size of the matrix: 1 3 5 2 3 2 4 6 1 3 2 4 1 4 2 5 3 6 1 1 2 2 1 4 7 2 5 8 3 6 9 1 2
Your turn Worked example Find (where possible): Find (where possible): 1 4 7 2 5 8 3 6 9 0 2 4 8 3 6 8 1 3 2 4 0 2 3 + + 4 7 0 1 0 3 1 0 0 0 0 1 7 8 9 1 3 5 2 4 6 1 3 2 4 1 4 2 5 3 6 + Not additively conformable 1 2 3 4
Your turn Worked example Find: Find: 1 3 5 2 4 6 51 2 4 7 3 7 14 28 42 21 25 71 2 5 3 6 4
Your turn Worked example Find the value of ?: Find the value of ?: 3 ? + ?2? ? 6 5 3? + ?2? 3? 20 = = 2? 2? ? =3 2
Your turn Worked example Write down the 4 4 identity matrix Write down the 2 2 identity matrix 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 Write down the 3 3 identity matrix
Chapter CONTENTS 6.2) Matrix multiplication
Your turn Worked example Determine the size of the matrix AB given the dimensions of A and B Determine the size of the matrix AB given the dimensions of A and B Dimensions of A Dimensions of B Dimensions of AB (if valid) Dimensions of A Dimensions of B Dimensions of AB (if valid) 3 2 2 1 2 3 3 1 2 1 3 2 2 4 1 4 1 4 Not valid 3 2 4 2 1 4 4 1 1 1 3 4 4 2 2 5 3 4 Not valid 2 4 4 2 3 3 3 3 3 3 2 4 2 4 2 2 2 4 2 2 2 2
Your turn Worked example Find the product of these matrices where possible: 1 2 3 4 5 6 Find the product of these matrices where possible: 1 2 3 4 5 7 6 8 7 8 9 10 11 12 27 61 95 30 68 106 33 75 117 1 3 5 2 4 6 7 9 8 10 1 4 2 5 3 6 7 9 8 10 7 9 8 10 1 4 2 5 3 6
Your turn Worked example Find: Find: 1 2 3 7 8 9 1 2 3 4 5 6 14 1 2 3 4 5 6 1 2 3 7 8 9 1 2 3 2 4 6 3 6 9
Your turn Worked example Find: Find: 4 2 1 3 2 4 1 3 2 4 199 435 290 634 3 1 3 2 4
Your turn Worked example Find: Find: ? 2 1 ? 0 1 1 0 ? 1 1 ?? 0 1 3 1 ? 0 1
Your turn Worked example 1 ? 1 1 2 0 ? 0 3 1 ? 2 0 1 0 3 ? = ? = 1 ? 4 9 4 3 1 1 8 6 7 6 9 7 6 4 Given that ?2= Given that ?2= , find the values , find the values 3 8 1 11 of ? and ? of ? and ? ? = 3,? = 2
Your turn Worked example 1 ? 2 ? ? = and ? = (? 2). Given that ?? = (0) ? = and ? = (1 ?). Given that ?? = (0) find ?? in terms of ? find ?? in terms of ? 2? 2?2 2 2? ?? =
Your turn Worked example Find: Find: 1 4 7 2 5 8 3 6 9 1 0 0 0 1 0 0 0 1 1 3 2 4 1 0 0 1 1 4 7 2 5 8 3 6 9 1 3 5 2 4 6 1 0 0 1 1 4 2 5 3 6 1 0 0 1 1 0 0 1 1 4 2 5 3 6
Chapter CONTENTS 6.3) Determinants
Your turn Worked example Calculate the determinant then decide if the matrix has an inverse. 1 0 Calculate the determinant then decide if the matrix has an inverse. 0 1 7 Yes 3 4 0 1 10 5 2 1 1 2 4 3 0 No 1 2 6 3
Your turn 4 1 Worked example 3 2 ? 1 4 ? ? + 2 3 ? ? = ? = Given that ? is singular, find the value of ?. Given that ? is singular, find the value of ?. ? =14 3
Your turn 1 2 4 5 1 8 Worked example 3 7 3 1 2 4 4 5 3 0 6 2 Find the minor of: Find the minor of: 1 0 2= 2 a) 2 a) 5 1 4 5 8= 37 b) -3 b) 0 1 1 2 8= 10 c) 7 c) -6 1
Your turn Worked example Calculate the determinant: Calculate the determinant: 2 5 8 1 4 0 1 4 2 5 8 54 0 6 2 6 2 1 1
Your turn 0 2 ? + 3 Worked example 1 4 2? + 1 3 1 0 where ? is a constant. Given that ? is singular, find the possible values of ? 2 3 ? 1 0 ? = ? = ? 1 2 5 where ? is a constant. Given that ? is singular, find the possible values of ? ? = 1 2, 9
Chapter CONTENTS 6.4) Inverting a 2 x 2 matrix
Your turn Worked example Find the inverse matrix for: Find the inverse matrix for 2 0 0 2 1 3 2 4 2 3 1 1 2 4 3 2 1 or 1 2 2 1 3 2 4 4 3 2 1
Your turn Worked example 1 2 ? ? + 3 4 ? + 2 3 ? For what value of ? is For what value of ? is 4 1 singular? singular? ? =14 3 Given ? is not this value, find the inverse. Given ? is not this value, find the inverse. 1 14 3? 1 3 ? ? + 2 4
Your turn Worked example If ? and ? are non-singular matrices, prove that ?? 1= ? 1? 1 If ? and ? are non-singular matrices, prove that ?? 1= ? 1? 1 Let ? = ?? 1 ?? ? = ?? ?? 1 ?? ? = ? ? 1??? = ? 1? ??? = ? 1 ?? = ? 1 ? 1?? = ? 1? 1 ?? = ? 1? 1 ? = ? 1? 1 ?? 1= ? 1? 1
Your turn Worked example If ? and ? are non-singular matrices such that ??? = ?, prove that ? = ? 1? ? If ? and ? are non-singular matrices such that ??? = ?, prove that ? = ? 1? ? ??? = ? ? 1??? = ? 1? ??? = ? 1 ?? = ? 1 ??? 1= ? 1? 1 ?? = ? 1? 1 ? = ? 1? 1
Chapter CONTENTS 6.5) Inverting a 3 x 3 matrix
Your turn 1 1 1 0 1 2 8 7 Worked example 3 2 4 2 3 4 0 1 1 0 2 3 4 , find ? 1. , find ? 1. If ? = If ? = 1 1 2 1 1 4
Your turn 3 0 2 Worked example 4 4 7 8 2 3 Show that ? 1= ?. 2 0 1 3 1 5 8 2 ? = ? = , , 1 Show that ? 1= ?. 2 0 1 3 1 3 0 2 2 0 1 3 1 3 0 2 1 0 0 0 1 0 0 0 1 ?2= = 1 1
Your turn 3 0 2 Worked example 4 4 7 8 2 3 The matrix ? is such that ?? 1= 2 5 3 4 1 8 1 0 11 Find ? 1. 2 0 1 3 1 5 8 2 ? = ? = , , 1 The matrix ? is such that ?? 1= 8 17 9 5 10 6 3 5 4 Find ? 1. . . ?? 1= ? 1? 1 ?? 1? = ? 1? 1? ?? 1? = ? 1 8 17 10 5 9 2 0 1 3 1 3 0 2 ? 1= 5 3 6 4 1 7 4 2 2 1 0 6 3 1 =
Your turn 1 0 2 Worked example ? 1 1 0 1 2 1 1 3 ? 1 3 1 ? = ,? 1 ? = ,? 1 1 1 Find the inverse matrix of A in terms of ? Find the inverse matrix of A in terms of ? 2 4 2 1 ? 3 2? 3 1 1 ? 1= ? + 1 1 2 2?
Your turn Worked example Find the inverse of the matrix using elementary row operations 3 2 0 Find the inverse of the matrix using elementary row operations 1 2 0 5 6 2 0 0 1 2 0 3 1 3 6 7 2 2 1 2 1 3 3 1
6.6) Solving systems of equations using matricesChapter CONTENTS
Worked example Your turn Solve the simultaneous equations: 6? ? + 2? = 6 ? + 2? 6? = 3 2? 3? 5? = 24 Solve the simultaneous equations: ? + 6? 2? = 21 6? 2? ? = 16 2? + 3? + 5? = 24 ? = 1,? = 4,? = 2
Your turn Worked example A colony of 1000 mole-rats is made up of adult males, adult females and youngsters. Originally there were 100 more adult females than adult males. After one year: The number of adult males had increased by 2% The number of adult females had increased by 3% The number of youngsters had decreased by 4% The total number of mole-rats had decreased by 20 A llama farmer has three types of llama: woolly, classic and Suri. Initially his flock had 2810 llamas in it. There were 160 more woolly llamas than classic. After one year: The number of woolly llamas had increased by 5% The number of classic llamas had increased by 3% The number of Suri llamas had decreased by 4% Overall the flock size had increased by 46 Form and solve a matrix equation to find out how many of each type of mole-rat were in the original colony. Form and solve a matrix equation to find out how many of each type of llama there were in the initial flock. 100 adult males, 200 adult females, 700 youngsters in the original colony
Your turn Worked example The system of equations is consistent and has a single solution. Determine the possible values of ?. 2? + 3? ? = 13 3? ? + ?? = 11 ? + ? + ? = 7 The system of equations is consistent and has a single solution. Determine the possible values of ?. ? 3? 2? = 7 ?? ? + 3? = 11 ? ? + ? = 13 ? 15
Your turn Worked example A system of equations is shown below: 3? ?? 6? = ? ?? + 3? + 3? = 2 3? ? + 3? = 2 For each of the following values of ?, determine whether the system of equations is consistent or inconsistent. If the system is consistent, determine whether there is a unique solution or an infinity of solutions. In each case, identify the geometric configuration of the plane corresponding to each value of ?. (a) ? = 0 (b) ? = 6 A system of equations is shown below: 3? ?? 6? = ? ?? + 3? + 3? = 2 3? ? + 3? = 2 For each of the following values of ?, determine whether the system of equations is consistent or inconsistent. If the system is consistent, determine whether there is a unique solution or an infinity of solutions. In each case, identify the geometric configuration of the plane corresponding to each value of ?. (a) ? = 1 3 1 1 3 1 6 3 3 (a) ? = 1: = 0 3 3? ? 6? = 1 ? + 3? + 3? = 2 3? ? + 3? = 2 1 + 2 2 :5? + 5? = 5 (4) 2 3 :4? + 4? = 4(5) Equations (4) and (5) are consistent so system is consistent and has an infinity of solutions. Planes meet at a sheaf (1) (2) (3)
Your turn Worked example A system of equations is shown below: ? ?? 6? = ? ?? 4? 12? = ? 3? + ?? + 18? = ? For each of the following values of ?,? and ?, determine whether the system of equations is consistent or inconsistent. If the system is consistent, determine whether there is a unique solution or an infinity of solutions. In each case, identify the corresponding geometric configuration. (a) ? = 2,? = 5,? = 4,? = 1 (b) ? = 2,? = 4,? = 6,? = 6 A system of equations is shown below: ? ?? 6? = ? ?? 4? 12? = ? 3? + ?? + 18? = ? For each of the following values of ?,? and ?, determine whether the system of equations is consistent or inconsistent. If the system is consistent, determine whether there is a unique solution or an infinity of solutions. In each case, identify the corresponding geometric configuration. (a) ? = 2,? = 4,? = 6,? = 5 1 2 2 4 6 6 12 18 = 0 (a) 3 ? 2? 6? = 2 2? 4? 12? = 4 3? + 6? + 18? = 5 All three planes are parallel and non-identical. The system of equations is inconsistent and has no solutions. (1) (2) (3)
