[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-448628-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-448628-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","411-the-inverse-and-determinant-of-a-matrix-practice-solutions","4.11 The Inverse and Determinant of a Matrix - Practice Solutions","","Practice solutions for AP Precalculus lesson 4.11 on determinants and matrix inverses. Includes multiple problem sets that ask for determinants and whether a matrix has an inverse, compute inverses when possible, verify inverse relationships by multiplying matrices, and calculate the area of parallelograms formed by vectors. The page also contains related test-prep questions involving exponential functions and piecewise-linear functions with graph sketching tasks.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/411-the-inverse-and-determinant-of-a-matrix-practice-solutions/448628/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/411-the-inverse-and-determinant-of-a-matrix-practice-solutions/448628.png","ImageObject",300,407,{"name":42,"@type":43},"Mabel","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-21","2026-09-30",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",6,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How do you determine whether a matrix has an inverse in this practice set?","Question",{"text":63,"@type":64},"Find the determinant for each matrix and use it to determine invertibility. 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Tell whether or not the matrix will have an inverse. |  |  |  |  |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| 1) !5 79& | 2) '\"! 3%&) |  |  |  | 3) !3.5 −96& |  |  |  |  |\n| Directions: Find the inverse of each matrix if possible. If it’s not possible, explain, why it isn’t. |  |  |  |  |  |  |  |  |  |\n| 4) !20 83& | 5) !−32 6& |  |  |  | 6) !5 8& |  |  |  |  |\n| CALCULATOR ACTIVE: Directions: Multiply the matrices and determine if they are inverses. |  |  |  |  |  |  |  |  |  |\n| 7) !53 84& ∙ 11 \\#%3\u003Cbr>| ⎡ − !' − ~~ ~~\\#~~ ~~ \" ⎤\u003Cbr>8) ⎢ \")&!!(! ~~ ~~!)!!\\#( ) )' ⎥ ∙ :32\u003Cbr>⎣ − ~~ ~~$~~ ~~ − $ & ⎦\u003Cbr>|  | 045 | 138; | 9) :001 | 010 | 100; ∙ :221 | −1 1 0 | 2; |\n| Directions: Plot the parallelogram formed by the vectors and then find the area. |  |  |  |  |  |  |  |  |  |\n| \u003Cbr>10) ⟨−3, 4⟩ and ⟨2, 5⟩ |  | \u003Cbr>11) ⟨4, 2⟩and ⟨3,−6⟩ |  |  |  |  |  |  |  |\n\n© The Algebros [from FlippedMath.com](from FlippedMath.com)  \n4.11 The Inverse and Determinant of a Matrix  4 . 11 Test Prep   \n12) (2.3)  \n\n| 􀀡 | 0 | 1 | 2 | 3 |  | 4 |  |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| 􀀧 (􀀡) | 40 \u003Cbr>| 20  | \u003Cbr>10 \u003Cbr>| | 5  | | !\u003Cbr>\" |\n\nThe exponential function 􀀫 is defined by 􀀫(􀀬) = 􀀮􀀯\\# , where 􀀮 and 􀀯 are positive constants. The table gives values of 􀀫 (􀀬) at selected values of 􀀬 . Which of the following statements is true?  \n(A) 􀀫 demonstrates exponential decay because 􀀮 > 0 and 0 \u003C 􀀯 \u003C 1.  \n(B) 􀀫 demonstrates exponential decay because 􀀮 > 0 and 􀀯 > 1.  \n(C) 􀀫 demonstrates exponential growth because 􀀮 > 0 and 0 \u003C 􀀯 \u003C 1.  \n(D) 􀀫 demonstrates exponential growth because 􀀮 > 0 and 􀀯 > 1.  \n13) (2.5A) Calculator active. In the 􀀬􀀲-plane, the graphs ofthe linear function 􀀳 and the exponential function 􀀴 both pass through the points (0, 4) and (1, 8) . The exponential function 􀀴 is in the form 􀀴 (􀀬) = 􀀮􀀯\\# . The function 􀀫 is given by 􀀫(􀀬) = 􀀴 (􀀬) − 􀀳(􀀬) . What is the minimum value of 􀀫 .  \n© The Algebros [from FlippedMath.com](from FlippedMath.com)  \n13) (2.7B) The piecewise-linear function 􀀫 , defined on −3 ≤ 􀀬 ≤ 3, is shown in the graph. The function 􀀹 is  \ngiven by 􀀹 (􀀬) = 􀀬 − 2. Sketch a graph of 􀀲 = 􀀫:􀀹 (􀀬); .  \n|  |  |  |  |  | | y |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n|  |  |  |  |  | 4 |  |  |  |  |  |  |\n|  |  |  | | | 3 |  |  |  |  |  |  |\n|  |  |  |  |  | |  |  |  |  |  |  |\n|  |  |  |  |  | 1 | |  |  |  |  | x |\n| - | 5 - | 4 - | 3 - | 2 - | 1 |  | | | 3 | 4 | 5 |\n|  |  |  |  |  | -1 |  |  |  |  |  |  |\n|  |  |  |  |  | -2 |  |  |  |  |  |  |\n\n1 2  \nGraph of 􀀧  \n© The Algebros [from FlippedMath.com](from FlippedMath.com)","cbCainCaYwXZ6EL6","https://ap.wps.com/l/cbCainCaYwXZ6EL6","pdf",1032728,"English","# 4.11 Practice Solutions\n## Determinants and Inverses\n## Calculator Active: Check Inverses by Multiplying\n## Parallelogram Area from Vectors","[{\"question\":\"How do you determine whether a matrix has an inverse in this practice set?\",\"answer\":\"Find the determinant for each matrix and use it to determine invertibility. The worksheet pairs “find the determinant” with “tell whether or not the matrix will have an inverse.”\"},{\"question\":\"What does the “Calculator active” instruction require for problems 7–9?\",\"answer\":\"Multiply the given matrices and determine if they are inverses based on the product results, as directed in the “Calculator active” section.\"},{\"question\":\"How are the area problems involving vectors handled?\",\"answer\":\"First plot the parallelogram formed by the vectors, then find its area, matching the directions for problems 10 and 11.\"}]","4.11 The Inverse and Determinant of a Matrix - Practice Solutions | PDF",1790728167]