Find the inverse of the following matrix if it exists. 1 −1 1 ������ = W0 1 −1X 1 3 −2 45
4. Properties of the Inverse Matrix: Let ������ and ������ be invertible matrices, ������ is a positive integer and ������ a non-zero scalar. Then the following statements are true. a) (������j#)j# = ������ b) s������tuj# = (������j#)t = ������jt c) (������������)j# = # ������j# d) (������w)j# = (������j#)w v e) (������������)j# = ������j#������j# March 2017 Q2(d) Consider the matrices ������ = U51 −02V and ������ = U14 −16V. i. Find ������j#. ii. Find matrix ������ such that ������������ = ������. 46
March 2016 Q2(d) Consider the matrices ������ = U21 −−43V and ������j# = U−21 11V. i. Find ������j#. ii. Find (2������������)j#. 47
5. Solving a System of Linear Equations by using Inverse • Write the system in the form ������������ = ������ 1 • Find ������j# 2 • Solve the system by using the formula ������ = ������j#������ 3 Solve the given system by using the inverse method. ������ + 2������ − ������ = 1 2������ + 5������ + ������ = −2 −������ − 2������ + 2������ = −1 48
Solve the following systems by using the inverse method. ������ + ������ + 2������ = −1 ������ + 2������ + 5������ = 0 ������ + 3������ + 7������ = 3 49
Solve the following systems by using the inverse method. ������ + ������ + ������ = 1 2������ + 4������ = −2 6������ − 5������ = 4 50
2.4 Elementary Matrices 1. An ������ × ������ matrix is called an elementary matrix if it can be obtained from the identity matrix ������) by performing a single elementary row operation. Given below are three elementary matrices and how they are obtained. Elementary Matrix Elementary Row Operation ������# = U10 20V ������& = U10 01V &{y⎯⎯z⎯→⎯y}z U01 02V = ������# 010 1 0 0 y{⎯~⎯↔⎯y}z 0 1 0 ������& = W1 0 0X ������' = W0 1 0X W1 0 0X = ������& 001 100 001 001 ������' = W0 1 3X 1 0 0 {yz⎯•⎯⎯'⎯y⎯€⎯↔⎯y}z 1 0 0 001 ������' = W0 1 0X W0 1 3X = ������' 001 001 Determine if each of the following matrices is an elementary matrix a) U01 03V 010 b) W1 0 0X 001 100 c) W0 1 2X 001 100 d) W0 0 0X 001 1 0 0 e) W0 2 0 X 0 0 −1 f) U04 0 00V 1 51
Determine which of the following matrices are elementary matrices. For elementary matrices, state the ERO used to obtain them. a) U− 12 01V b) U− 02 11V c) U10 20V d) •01 1‚04ƒ 100 e) W0 0 1X 010 150 f) W0 1 0X 001 52
2. Theorem: Let ������ be the elementary matrix obtained by performing an elementary row operation on ������1. If the same elementary row operation is performed on an ������ × ������ matrix ������, then the resulting matrix is given by the product ������������. If ������1×) {„⎯y}… ������1×) and ������)×) {†‡⎯1⎯⎯ˆ⎯ ⎯„⎯y}… ������ then ������ = ������������ Reduce ������ using ERO. ������ = U01 2 34V y{⎯~⎯↔⎯y}z U10 1 34V = ������ Find elementary matrix. 1 2 ������& = U10 01V {y⎯~⎯↔⎯y}z U01 01V = ������# The matrix ������ can be obtained as ������ = ������������ ������������ = Consider the following matrices. 1 2 3 001 ������ = W0 1 4 X , ������ = W0 1 0X 2 1 −1 100 We can obtain E from ������' as follows. 1 0 0 {y⎯~⎯↔⎯y}€ 0 0 1 ������' = W0 1 0X W0 1 0X = ������ 001 100 If we perform the same elementary operation on ������, we get Based on the theorem, the result is the same as finding ������������. 53
From the previous example, the theorem can also be applied onto the following matrices. ������ = U13 2 0 21V , ������ = U10 20V 1 4 54
Find the sequence of elementary matrices that can be used to rewrite the matrix ������ in its row echelon form. 2 4 6 ������ = W0 −1 3 X 3 6 10 55
3. Row-Equivalence: Let ������ and ������ be ������ × ������ matrices. Matrix ������ is row equivalent to ������ if there exists a finite number of elementary matrices such that ������ = ������t������tj#. . . ������&������#������ Consider the following matrices. −2 0 −1 −1 −2 0 −1 −1 ������ = W 6 2 1 3X and ������ = W 0 2 −2 0X 0 2 −2 0 0 0 0 0 Find elementary matrices ������# and ������& such that ������ = ������&������#������. 56
Consider the following matrices. 2 4 −1 −4 1 0 ������ = W−4 1 0 X and ������ = W 0 −2 −5X 1 3 2 1 3 2 Find elementary matrices ������# and ������& such that ������ = ������&������#������. 57
Find the sequence of elementary matrices that can be used to rewrite the matrix ������ in its row echelon form. 2 4 6 ������ = W0 −1 3 X 3 6 10 58
4. If A is an invertible matrix, then there are elementary matrices ������#, ������&, … , ������tj#, ������t such that ������ = ������t������tj#. . . ������&������#������ Consider the following matrix. ������ = U− 15 20V Find elementary matrices ������# and ������& such that ������&������#������ = ������&. 59
Consider the following matrix. 010 ������ = W1 0 0X 201 Find elementary matrices ������# and ������& such that ������&������#������ = ������'. 60
Consider the following matrix. 030 ������ = W1 0 0X 401 Find elementary matrices ������#, ������& and ������' such that ������'������&������#������ = ������'. 61
5. Inverse Elementary Matrix ������j������: Every elementary matrix ������ has an inverse, ������j# that is also an elementary matrix. 6. The inverse elementary matrix is obtained by performing the inverse of elementary row operation that is used to produce ������ from ������). Elementary row Inverse of elementary operation row operation ������) ������ ������) ������) ������j# ������) Elementary Row Inverse of Elementary Row Operation Operation ������2 ↔ ������3 ������2 ↔ ������3 ������������2 → ������2 1 ������2 → ������2 ������2 + ������������3 → ������2 ������ ������2 + (−������)������3 → ������2 7. If A is an invertible matrix, then A can be written as a product of inverse elementary matrices������#, ������&, … , ������tj#, ������t such that ������ = (������#)j#(������&)j# ∙ ∙ ∙ (������tj#)j#(������t)j# 62
Consider the following matrices. 1 3 1 4X and ������ = W−3 12X 3 2 0 0 ������ = W−1 6 a) Find elementary matrices ������# and ������& such that ������&������#������ = ������'. b) Find the inverse elementary matrices for ������#, and ������& in (a). 63
c) Write ������ as a product of elementary matrices. d) Write ������j# as a product of elementary matrices. 64
CHAPTER 3: DETERMINANTS 3.1 Introduction to Determinants 1. A function that outputs a real value number of a square (������ × ������) matrix. 2. Determinant of ������ is denoted by |������| or det(������). 3. Evaluating determinant of ������ × ������ matrix If ������ = (������..), then |������| = ������.. Example 3.1 Find the determinants of the following matrices. a) ������ = (−5) b) ������ = (8) c) C = (−76) 4. Evaluating determinant of ������ × ������ matrix ONLY If ������ = 7������������.8.. ������������.8889, then |������| = ������..������88 − ������.8������8. Or If ������ = 7������������ ������������9, then |������| = ������������ − ������������ Example 3.2 Find the determinants of the following matrices. a) ������ = 7− 31 429 b) ������ = 701 − 329 65
5. Evaluating determinant of ������ × ������ matrix ONLY using Sarrus’s scheme. • Copy the matrix and rewrite the first two columns on the right side of the matrix. ������.. ������.8 ������.E ������.. ������.8 ������ = D������8. ������88 ������8EF ������8. ������88 ������E. ������E8 ������EE ������E. ������E8 Then |������| = (������..������88������EE + ������.8������8E������E. + ������.E������8.������E8) − (������.E������88������E. + ������..������8E������E8 + ������.8������8.������EE) WARNING!! This method is for ������ × ������ matrix ONLY! Example 3.3 Evaluate the determinant of the following matrices using Sarrus’s scheme. 1 −2 3 a) H0 1 4H 521 −1 0 1 b) H 2 3 −4H −1 1 0 ������ 1 ������ c) H1 0 1H ������ ������ 0 66
3.2 Evaluating Determinants by using Cofactor Expansion 1. Minor J������������������N: If ������ is a square matrix with elements ������OP, the minor ������������������ of the element ������OP is defined as the determinant of the matrix obtained by deleting the ������������������ row and the ������������������ column. Example 3.4 123 Let ������ = D4 5 6F. Find ������.8 and ������8.. 789 2. Cofactor J������������������N: The cofactor, ������OP for the elements ������OP is defined by where (−1)OWP = X+−11 ������������������ = (−������)������W������������������������ if ������ + ������ = even if ������ + ������ = odd • The effect of the positive and negative sign of the cofactor can be represented as matrix below. ������.. ������.8 ������.E ⋯ + − + ⋯ [������������8E.. ������������8E88 ������������8EEE ⋯⋯^ [−+ + − ⋯⋯^ − + ⋮ ⋮ ⋮ ⋮ ⋮⋮⋮⋮ Example 3.5 123 Let ������ = D4 5 6F. Find ������.8 and ������8.. 789 67
3. Cofactor Expansion: The determinant of an ������ × ������ matrix ������ can be evaluated by multiplying every element along a certain row (or column) with the value of its cofactor and then adding up all the products. a |������| = _ ������.P������.P = ������..������.. + ������.8������.8 + ������.E������.E + . . . + ������.a������.a Pb. • This is the cofactor expansion along the first row. • TIPS! q You can choose any row or column for the cofactor expansion. q Choose the row or column with the MOST zeros for simpler calculation. Example 3.6 2 −3 5 Find the determinants of the matrix ������ = D 1 4 3F by using cofactor expansion along −2 −1 6 a) the first row b) the second row 68
Example 3.7 2 1 2 −1 Find the determinants of the matrix ������ = c− 04 1 0 − 32d by using cofactor expansion. 3 0 0 1 0 6 69
Example 3.8 3 2 6 Given matrix ������ = D−1 4 7F. Find all the minors and cofactors. 5 1 −2 70
Example 3.9 Evaluate the determinants of the following matrices by using cofactor expansion. 1 4 2 a) D0 6 −3F 1 −1 7 −2 3 1 b) D 1 −5 −1F 4 2 1 71
1 2 0 4 c) c05 −1 −2 − 01d 1 0 0 −2 4 0 72
3.3 Evaluating Determinants by using Row or Column Operations 1. Interchange rows If ������ ie⎯f↔⎯⎯ekh ������ or ������ ilf⎯↔⎯klh ������ then |������| = −|������| or |������| = −|������|. Example 3.10 3 −2 4 Given ������ = D5 6 −7F and |������| = 96. 10 4 10 4 If ������ = D5 6 −7F, then |������| = 3 −2 4 −2 3 4 If ������ = D 6 5 −7F, then |������| = 014 73
2. Multiply a row with ������ If ������ nie⎯⎯f→⎯kef ������ or ������ inl⎯f⎯↔⎯klf ������ then |������| = ������|������| or |������| = ������ |������|. ������ Example 3.11 3 −2 4 Given ������ = D5 6 −7F and |������| = 96. 10 4 6 −4 8 If ������ = D5 6 −7F, then |������| = 10 4 3 −2 −2 If ������ = D5 6 3.5F, then |������| = 1 0 −2 74
3. Add a row, which is multiplied by ������, to another row If ������ eif⎯W⎯n⎯e⎯h⎯→⎯kef ������ or ������ ilf⎯W⎯n⎯l⎯h⎯↔⎯klf ������ then |������| = |������| or |������| = |������|. Example 3.12 3 −2 4 Given ������ = D5 6 −7F and |������| = 96. 10 4 0 −2 −8 If ������ = D5 6 −7F, then |������| = 10 4 30 4 p If ������ = [5 8 −7^, then |������| = 12 4 75
Example 3.13 Find the determinant of the following matrix using elementary row (or column) operations. a) q75 31q 6 8 −7 b) H1 4 2 H 1 3 −2 3 −2 4 5 c) r10 3 −4 1 8 12r 2 −4 16 6 76
Example 3.14 ������ ������ ������ 6������ + ������ 6������ + ������ 6������ + ℎ If H������ ������ ������H = −2, find the value of x������ − 2������ ������ − 2������ ������ − 2ℎx. ������ ℎ ������ −3������ −3������ −3ℎ 77
Example 3.15 ������ ������ ������ 2������ + ������ ������ −2������ If H������ ������ ������H = −2, find the value of H−10������ − 5������ −5������ 10������H. ������ ℎ ������ 2ℎ + ������ ℎ −2������ 78
3.4 Properties of Determinants 1. If ������ and ������ are square matrix of size ������ × ������, then |������������| = |������||������| Example 3.16 Let ������ = 714 −−219 and ������ = 742 − 019, find a) |������������| b) |������||������| c) |������������| 79
2. If ������ is a square matrix of size ������ × ������ and ������ is a positive integer, then z������nz = |������|n Example 3.17 Let ������ = 714 −−129, find a) |������8| b) |������|8 3. If ������ is a square matrix of size ������ × ������ and ������ is a scalar, then |������������| = ������a|������| Example 3.18 Let ������ = 7− 22 − 319, find |3������|. 80
Example 3.19 If ������E×E and ������E×E have determinants |������| = −2 and |������| = 5, find a) |������������| b) |������E| c) |(������������)8| d) |������������������������������������| e) |������{������E| f) | − 4������| g) |5������������| h) |(2������)8| 81
4. If ������ is a non-singular, then the determinant of the inverse of ������ is |������|.| = 1 |������| Example 3.20 Let ������ = 7− 22 − 319, find |������|.|. 5. If ������ is a square matrix of size ������ × ������, then the determinant of the transpose of ������ is |������}| = |������| Example 3.21 Let ������ = 7− 42 − 319, find |������}|. 82
Example 3.22 Given ������E×E and ������E×E with determinants |������| = 3 and |������| = 2, find a) |2������}| b) |������|.������8| c) |(2������)|.| d) |3������}������|.| 83
Example 3.23 a) Given ������p×p and ������p×p with |������|.������}| = −2 and |������| = 3, find |������|. b) Given ������8×8 and ������8×8 with |4(������������)|.| = 2 and |������| = −1, find |������|. 84
3.5 Applications of Determinants 1. Cramer’s Rule: A method to solve system of ������ linear equations with ������ variables. Consider the system in form of ������������ = ������. ������.. ������.8 ⋯ ������.a ������. ������. ������8. ������88 ⋯ ������8a c������⋮8 d c������⋮8 c ⋮ ⋱ ⋮ d = d ⋮ ������a. ������a8 ⋯ ������aa ������a ������a Step 1: Calculate det(������) ������.. ������.8 ⋯ ������.a ������8. ������88 ⋯ ������8a |������| = r ⋮ ⋮ ⋱ ⋮ r ������a. ������a8 ⋯ ������aa Step 2: Calculate det(������€•) (Substitute the ������. column with ������). ������������ ������.8 ⋯ ������.a z������€•z = r������⋮������ ������88 ⋯ ������8a ⋮ r ⋮ ⋱ ������������ ������a8 ⋯ ������aa Step 3: Calculate ������. ������. = z������€• z |������| • To solve for the variable ������a, substitute the ������a column with ������. 85
Example 3.24 Solve the following system of linear equations. 2������ − ������ + ������ = −1 3������ + ������ − 2������ = 0 ������ + 2������ + ������ = 3 86
Example 3.25 Solve the following system of linear equations. ������ + ������ − ������ = 1 2������ + 3������ + ������������ = 3 (������ − 1)������ + 4������ = 1 87
2. Adjoint Method: A method to find the inverse of a matrix. • Alternative way to calculate the inverse of a matrix, ������|.. ������|. = 1 × ������������������(������) |������| where ������������������(������) = [������(������)] = ˆ������OP‰} ������|. = 1 × ˆ������OP ‰} |������| ˆ������OP‰ is the cofactor matrix Example 3.26 Find the inverse of ������ = 757 129 using the adjoint method. 88
Example 3.27 2 −1 −3 Find the inverse of ������ = D 1 0 2 F using the adjoint method. −1 1 4 89
Example 3.28 −1 1 2 −8 0 ������ Given the cofactor matrix for ������ = D−1 0 4F is ������Š = D ������ 0 2 F, find 0 20 4 21 a) the values of ������ and ������. b) the inverse matrix of ������. 90
CHAPTER 4: VECTOR SPACES 4.1 Vectors in ������������ Vector Magnitude A vector represents a quantity that has Quantity that has only a magnitude is both magnitude (distance) and direction. called a scalar. • Ali's car is traveling southwest at 65 • Ali's car is traveling at 65 miles per miles per hour hour 1. Vector notation • A small letter in boldface (in print) – v, u, a • An arrow over a letter (handwritten) – ���⃗���, ���&⃗���, ���⃗��� • An arrow over two letters – &���&���&&���&⃗���, ���&&���&&���&⃗��� Terminal point • ������(������>, ������>) ������ = &���&���&&���&⃗��� = ������ − ������ • ������(������=, ������=) Initial point Vectors in a Plane .������������0 Vectors in a Space .������������0 Presented geometrically by a directed (Geometrically) Begins from the origin line segment begins from the origin and and ends at the point (������, ������, ������). ends at the point (������, ������). Written as ������ = (������, ������) or ������ = 9������������:. ������ Written as ������ = (������, ������, ������) or ������ = ;������<. ������ 91
������ ������ Terminal point ������(0,0,0) Terminal point • ������(������, ������) Initial point • ������(������, ������, ������) ������ = (������, ������) ������ = (������, ������, ������) ������ ������(0,0) ������ Initial point ������ Vector ������ in ������> Vector in ������ in ������A 2. Vectors in ������-space (������ ordered tuples) If ������ is a positive integer, then An ordered ������-tuple is a sequence of ������ real numbers .������������, ������������, . . . , ������������0. The set of all ������-ordered tuples is called the ������-space and denoted by ������������. • ������������ = 1-dimensional real coordinate space (set of all real numbers) • ������������ = 2-dimensional real coordinate space (set of all ordered pair of real numbers (������, ������)) • ������������ = 3-dimensional real coordinate space (set of all ordered triple of real numbers (������, ������, ������)) • ������������ = 4-dimensional real coordinate space (set of all ordered quadruple of real numbers (������=, ������>, ������A, ������G)) 3. Zero vector: A zero vector denoted by 0 or 0&⃗, is a vector of magnitude 0 and thus has all components equal to zero. ������ ������ zero vector vector 4. Negative vector: A vector having the same magnitude but opposite direction to a vector ������ is −������. ������ −������ negative vector 92
5. Equal vectors: Two vectors are equal if they have the same magnitude and direction. ������ ������ ������ = ������ Equal vectors have the same length and direction but may have different starting points. 6. Scalar Multiplication: Let ������ = (������=, ������>, … , ������K) be a vector in ������K and ������ be a scalar, then ������������ = (������������=, ������������>, … , ������������K) 2������ Notes: ������ • If ������ = ������������, then ������ and ������ are said to be parallel. • If ������ > ������, ������ and ������ have the same direction. − 1 ������ • If ������ < ������, ������ and ������ are in the opposite direction. 2 Example 4.1 b) − 1 ������ 3 If ������ = (1, −3), find a) 2������ 93
7. Vector addition/ subtraction: Let ������ = (������=, ������>, . . . , ������K) and ������ = (������=, ������>, . . . , ������K) be any vectors in ������K, then ������ + ������ = (������=, ������>, . . . , ������K) + (������=, ������>, . . . , ������K) = (������= + ������=, ������> + ������>, . . . , ������K + ������K) ������ + (−������) = (������=, ������>, . . . , ������K) − (������=, ������>, . . . , ������K) = (������= − ������=, ������> − ������>, . . . , ������K − ������K) Example 4.2 b) 2������ − ������ If ������ = (2, − 1, 8) and ������ = (−1, 3, 2), find a) ������ + ������ 94
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