{"id":103563,"date":"2022-05-14T12:14:37","date_gmt":"2022-05-14T06:44:37","guid":{"rendered":"https:\/\/ccssanswers.com\/?p=103563"},"modified":"2022-05-10T14:22:57","modified_gmt":"2022-05-10T08:52:57","slug":"eureka-math-precalculus-module-2-lesson-11","status":"publish","type":"post","link":"https:\/\/ccssanswers.com\/eureka-math-precalculus-module-2-lesson-11\/","title":{"rendered":"Eureka Math Precalculus Module 2 Lesson 11 Answer Key"},"content":{"rendered":"

Engage NY Eureka Math Precalculus Module 2 Lesson 11 Answer Key<\/h2>\n

Eureka Math Precalculus Module 2 Lesson 11 Exercise Answer Key<\/h3>\n

Opening Exercise
\nKiamba thinks A+B=B+A for all 2\u00d72 matrices. Rachel thinks it is not always true. Who is correct? Explain.
\nAnswer:
\nLet A=\\(\\left(\\begin{array}{ll}
\na_{11} & a_{12} \\\\
\na_{21} & a_{22}
\n\\end{array}\\right)\\) and B=\\(\\left(\\begin{array}{ll}
\n\\boldsymbol{b}_{11} & \\boldsymbol{b}_{12} \\\\
\n\\boldsymbol{b}_{21} & \\boldsymbol{b}_{22}
\n\\end{array}\\right)\\)
\nWhat is the sum of A+B?
\n\"Eureka
\nThe two matrices must be equal because each of the sums must be equal according to the commutative property of addition of real numbers. Kiamba is correct.<\/p>\n

Exercises<\/p>\n

Exercise 1.
\nIn two-dimensional space, let A be the matrix representing a rotation about the origin through an angle of 45\u00b0, and let B be the matrix representing a reflection about the x-axis. Let x be the point \\(\\left(\\begin{array}{l}
\n1 \\\\
\n1
\n\\end{array}\\right)\\).
\n\"Eureka<\/p>\n

a. Write down the matrices A, B, and A+B.
\nAnswer:
\n\"Eureka<\/p>\n

b. Write down the image points of Ax, Bx, and (A+B)x, and plot them on graph paper.
\nAnswer:
\nAx=\\(\\left(\\begin{array}{c}
\n\\mathbf{0} \\\\
\n\\sqrt{2}
\n\\end{array}\\right)\\) Bx=\\(\\left(\\begin{array}{c}
\n1 \\\\
\n-1
\n\\end{array}\\right)\\) (A+B)x=\\(\\left(\\begin{array}{c}
\n1 \\\\
\n\\sqrt{2}-1
\n\\end{array}\\right)\\)<\/p>\n

c. What do you notice about (A+B)x compared to Ax and Bx?
\nAnswer:
\nThe point (A+B)x is equal to the sum of the points Ax and Bx by the distributive property.<\/p>\n

Exercise 2.
\nFor three matrices of equal size, A, B, and C, does it follow that A+(B+C)=(A+B)+C?
\na. Determine if the statement is true geometrically. Let A be the matrix representing a reflection across the y-axis. Let B be the matrix representing a counterclockwise rotation of 30\u00b0. Let C be the matrix representing a reflection about the x-axis. Let x be the point \\(\\left(\\begin{array}{l}
\n1 \\\\
\n1
\n\\end{array}\\right)\\).
\n\"Eureka
\nAnswer:
\n\"Eureka
\nFrom the graph, we see that Ax+(Bx+Cx)=(Ax+Bx)+Cx.<\/p>\n

b. Confirm your results algebraically.
\nAnswer:
\n\"Eureka<\/p>\n

c. What do your results say about matrix addition?
\nAnswer:
\nMatrix addition is associative.<\/p>\n

Exercise 3.
\nIf x=\\(\\left(\\begin{array}{l}
\n\\boldsymbol{x} \\\\
\n\\boldsymbol{y} \\\\
\n\\boldsymbol{Z}
\n\\end{array}\\right)\\), what are the coordinates of a point y with the property x+y if the origin O=\\(\\left(\\begin{array}{l}
\n0 \\\\
\n0 \\\\
\n0
\n\\end{array}\\right) ?\\)
\nAnswer:
\ny=\\(\\left(\\begin{array}{c}
\n-x \\\\
\n-y \\\\
\n-z
\n\\end{array}\\right)\\)<\/p>\n

Exercise 4.
\nSuppose A=\\(\\left(\\begin{array}{ccc}
\n11 & -5 & 2 \\\\
\n-34 & 6 & 19 \\\\
\n8 & -542 & 0
\n\\end{array}\\right)\\) and matrix B has the property that Ax+Bx is the origin. What is the matrix B?
\nAnswer:
\nB=\\(\\left(\\begin{array}{ccc}
\n-11 & 5 & -2 \\\\
\n34 & -6 & -19 \\\\
\n-8 & 542 & 0
\n\\end{array}\\right)\\)<\/p>\n

Exercise 5.
\nFor three matrices of equal size, A, B, and C, where A represents a reflection across the line y=x, B represents a counterclockwise rotation of 45\u00b0, C represents a reflection across the y-axis, and x=\\(\\left(\\begin{array}{l}
\n1 \\\\
\n2
\n\\end{array}\\right)\\):
\na. Show that matrix addition is commutative: Ax+Bx=Bx+Ax.
\nAnswer:
\n\"Eureka<\/p>\n

b. Show that matrix addition is associative: Ax+(Bx+Cx)=(Ax+Bx)+Cx.
\nAnswer:
\n\"Eureka<\/p>\n

Exercise 6.
\nLet A, B, C, and D be matrices of the same dimensions. Use the commutative property of addition of two matrices to prove A+B+C=C+B+A.
\nAnswer:
\nIf we treat A+B as one matrix, then:
\n(A+B)+C=C+(A+B)
\n=C+(B+A)
\n=C+B+A<\/p>\n

Eureka Math Precalculus Module 2 Lesson 11 Problem Set Answer Key<\/h3>\n

Question 1.
\nLet A be a matrix transformation representing a rotation of 45\u00b0 about the origin and B be a reflection across the y-axis. Let x=(3,4).
\na. Represent A and B as matrices, and find A+B.
\nAnswer:
\n\"Eureka<\/p>\n

b. Represent Ax and Bx as matrices, and find (A+B)x.
\nAnswer:
\n\"Eureka<\/p>\n

c. Graph your answer to part (b).
\nAnswer:
\nSee graph in part (d).<\/p>\n

d. Draw the parallelogram containing Ax, Bx, and (A+B)x.
\nAnswer:
\n\"Eureka<\/p>\n

Question 2.
\nLet A be a matrix transformation representing a rotation of 300\u00b0 about the origin and B be a reflection across the
\nx-axis. Let x=(2,-5).
\na. Represent A and B as matrices, and find A+B.
\nAnswer:
\n\"Eureka<\/p>\n

b. Represent Ax and Bx as matrices, and find (A+B)x.
\nAnswer:
\n\"Eureka<\/p>\n

c. Graph your answer to part (b).
\nAnswer:
\nSee graph in part (d).<\/p>\n

d. Draw the parallelogram containing Ax, Bx, and (A+B)x.
\nAnswer:
\n\"Eureka<\/p>\n

Question 3.
\nLet A, B, C, and D be matrices of the same dimensions.
\na. Use the associative property of addition for three matrices to prove (A+B)+(C+D)=A+(B+C)+D.
\nAnswer:
\nIf we treat C+D as one matrix, then:
\n(A+B)+(C+D)=
\n=A+(B+(C+D))
\n=A+((B+C)+D)
\n=A+(B+C)+D<\/p>\n

b. Make an argument for the associative and commutative properties of addition of matrices to be true for finitely many matrices being added.
\nAnswer:
\nFor finitely many matrices, we can always use the formula that has been proven already and break the rest of the problem down into that many pieces, just like we did in part (a). Since this is true for any finite number, we could also use the formula that has been proven for one less than whatever number we are trying to prove the property true for.<\/p>\n

Question 4.
\nLet A be a m\u00d7n matrix with element in the ith<\/sup> row, jth<\/sup> column aij<\/sub>, and B be a m\u00d7n matrix with element in the ith<\/sup> row, jth<\/sup> column bij<\/sub>. Present an argument that A+B=B+A.
\nAnswer:
\nThe entry in the ith<\/sup> row, jth<\/sup> column of A+B will be aij<\/sub>+bij<\/sub>, which is equal to bij<\/sub>+aij<\/sub>, which is the entry in the ith<\/sup> row, jth<\/sup> column of B+A. Since this is true for any element of A+B, we have that A+B=B+A for any two matrices of equal dimensions.<\/p>\n

Question 5.
\nFor integers x, y, define x\u2295y=x\u22c5y+1; read \u201cx plus y\u201d where x\u22c5y is defined normally.
\na. Is this form of addition commutative? Explain why or why not.
\nAnswer:
\nYes x\u2295y=xy+1=yx+1=y\u2295x<\/p>\n

b. Is this form of addition associative? Explain why or why not.
\nAnswer:
\nNo
\nx\u2295(y\u2295z)=x\u2295(yz+1)=x(yz+1)+1=xyz+x+1
\n(x\u2295y)\u2295z=(xy+1)\u2295z=(xy+1)z+1=xyz+z+1<\/p>\n

Question 6.
\nFor integers x, y, define x\u2295y=x.
\na. Is this form of addition commutative? Explain why or why not.
\nAnswer:
\nNo x\u2295y=x, and y\u2295x=y<\/p>\n

b. Is this form of addition associative? Explain why or why not.
\nAnswer:
\nYes x\u2295(y\u2295z)=x\u2295y=x, and (x\u2295y)\u2295z=x\u2295z=x<\/p>\n

Eureka Math Precalculus Module 2 Lesson 11 Exit Ticket Answer Key<\/h3>\n

Question 1.
\nLet x=\\(\\left(\\begin{array}{l}
\n3 \\\\
\n-1
\n\\end{array}\\right)\\), A=\\(\\left(\\begin{array}{ll}
\n2 & 2 \\\\
\n2 & 0
\n\\end{array}\\right)\\), and B=\\(\\left(\\begin{array}{cc}
\n-1 & 0 \\\\
\n0 & -1
\n\\end{array}\\right)\\).
\na. Find and plot the points Ax, Bx, and (A+B)x on the axes below.
\n\"Eureka
\nAnswer:
\n\"Eureka<\/p>\n

b. Show algebraically that matrix addition is commutative: Ax+Bx=Bx+Ax.
\nAnswer:
\nAx=\\(\\left(\\begin{array}{l}
\n4 \\\\
\n6
\n\\end{array}\\right)\\), Bx=\\(\\left(\\begin{array}{l}
\n-3 \\\\
\n1
\n\\end{array}\\right)\\), Ax+Bx=\\(\\left(\\begin{array}{l}
\n1 \\\\
\n7
\n\\end{array}\\right)\\), and Bx+Ax=\\(\\left(\\begin{array}{l}
\n1 \\\\
\n7
\n\\end{array}\\right)\\) Ax+Bx=Bx+Ax<\/p>\n

Question 2.
\nLet A=\\(\\left(\\begin{array}{ll}
\na & b \\\\
\nc & d
\n\\end{array}\\right)\\) and B=\\(\\left(\\begin{array}{ll}
\nx & y \\\\
\nz & w
\n\\end{array}\\right)\\). Prove A+B=B+A.
\nAnswer:
\n\"Eureka<\/p>\n","protected":false},"excerpt":{"rendered":"

Engage NY Eureka Math Precalculus Module 2 Lesson 11 Answer Key Eureka Math Precalculus Module 2 Lesson 11 Exercise Answer Key Opening Exercise Kiamba thinks A+B=B+A for all 2\u00d72 matrices. Rachel thinks it is not always true. Who is correct? Explain. Answer: Let A= and B= What is the sum of A+B? The two matrices … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/103563"}],"collection":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/comments?post=103563"}],"version-history":[{"count":1,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/103563\/revisions"}],"predecessor-version":[{"id":136157,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/103563\/revisions\/136157"}],"wp:attachment":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/media?parent=103563"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/categories?post=103563"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/tags?post=103563"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}