[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-661655-105":59,"doc-detail-661655-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","linear-equations-in-linear-algebra-chapter-1-counterexample-on-unique-echelon-form","Linear Equations in Linear Algebra - Chapter 1 - Counterexample on Unique Echelon Form","","Solutions-style content addressing a key claim in linear algebra: that every matrix is row equivalent to a unique matrix in echelon form. The text explains a counterexample strategy using a starting matrix A, transforming it by different sequences of row operations into two distinct echelon-form matrices X and Y. It concludes that both X and Y are row equivalent to A, so echelon-form equivalents need not be unique. The material references Chapter 1 and nearby sections/exercises.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/linear-equations-in-linear-algebra-chapter-1-counterexample-on-unique-echelon-form/661655/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/linear-equations-in-linear-algebra-chapter-1-counterexample-on-unique-echelon-form/661655.png","ImageObject",300,407,{"name":92,"@type":93},"Blitz","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-28",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why is the statement about a unique row-equivalent echelon form false?","Question",{"text":112,"@type":113},"The text shows that a single matrix A can be transformed by different row-operation sequences into two different echelon-form matrices X and Y.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are matrices X and Y obtained in the counterexample?",{"text":117,"@type":113},"Starting from A, one applies row operations to reach X in echelon form, then applies another set of row operations to reach Y in echelon form.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the key conclusion about echelon-form matrices that are row equivalent to the same matrix?",{"text":121,"@type":113},"Multiple echelon-form matrices can be row equivalent to a given matrix, so uniqueness does not hold.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},661655,1793173556,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":14,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":24},24464137899374,"https://us-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Linear algebra and its applications 6th edition solutions manual pdf  \nThe statement that every matrix is row equivalent to a unique matrix in echelon form is false. This can be demonstrated with a counterexample, where two different matrices (X and Y) are shown to be row equivalent to the same matrix in echelon form. To illustrate this, consider the following steps: 1 . Start with a given matrix A. 2. Apply row operations to transform A into matrix X, which is in echelon form. 3. Then, apply another set of row operations to transform A into matrix Y, which is also in echelon form. By doing so, it can be shown that matrices X and Y are both in echelon form and are row equivalent to the original matrix A. This means that not every matrix has a unique equivalent matrix in echelon form. In essence, this shows that the statement \"every matrix is row equivalent to a unique matrix in echelon form\" is false, as there can be multiple matrices in echelon form that are row equivalent to a given matrix. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 Chapter 1: Linear Equations in Linear Algebra (cont'd 3) Section 1.8 Section 1.9 Exercise 1 Exercise 9 Exercise 17 Exercise 25 Exercise 33 Exercise 1 Exercise 9 Exercise 17 Exercise 25 Exercise 33 Exercise 2 Exercise 10 Exercise 18 Exercise 26 Exercise 34 Exercise 2 Exercise 10 Exercise 18 Exercise 26 Exercise 34 Exercise 3 Exercise 11 Exercise 19 Exercise 27 Exercise 35 Exercise 3 Exercise 11 Exercise 19 Exercise 27 Exercise 35 Exercise 4 Exercise 12 Exercise 20 Exercise 28 Exercise 36 Exercise 4 Exercise 12 Exercise 20 Exercise 28 Exercise 36 Exercise 5 Exercise 13 Exercise 21 Exercise 29 Exercise 37 Exercise 5 Exercise 13 Exercise 21 Exercise 29 Exercise 37 Exercise 6 Exercise 14 Exercise 22 Exercise 30 Exercise 38 Exercise 6 Exercise 14 Exercise 22 Exercise 30 Exercise 38 Exercise 7 Exercise 15 Exercise 23 Exercise 31 Exercise 39 Exercise 7 Exercise 15 Exercise 23 Exercise 31 Exercise 39 Exercise 8 Exercise 16 Exercise 24 Exercise 32 Exercise 40 Exercise 8 Exercise 16 Exercise 24 Exercise 32 Exercise 40 Chapter 1: Linear Equations in Linear Algebra (cont'd 4) Section 1.10 Supp. Exercises Exercise 1 Exercise 6 Exercise 11 Exercise 1 Exercise 6 Exercise 11 Exercise 16 Exercise 21 Exercise 2 Exercise 7 Exercise 12 Exercise 2 Exercise 7 Exercise 12 Exercise 17 Exercise 22 Exercise 3 Exercise 8 Exercise 13 Exercise 3 Exercise 8 Exercise 13 Exercise 18 Exercise 23 Exercise 4 Exercise 9 Exercise 14 Exercise 4 Exercise 9 Exercise 14 Exercise 19 Exercise 24 Exercise 5 Exercise 10 Exercise 5 Exercise 10 Exercise 15 Exercise 20 Exercise 25 Back to Top Top","cbCaigAPlg5ZlsKP","https://ap.wps.com/l/cbCaigAPlg5ZlsKP","pdf",38532,"English","# Chapter 1: Linear Equations in Linear Algebra (cont'd)\n## Section 1.8 - Section 1.9\n## Section 1.10\n## Supplementary Exercises","[{\"question\":\"Why is the statement about a unique row-equivalent echelon form false?\",\"answer\":\"The text shows that a single matrix A can be transformed by different row-operation sequences into two different echelon-form matrices X and Y.\"},{\"question\":\"How are matrices X and Y obtained in the counterexample?\",\"answer\":\"Starting from A, one applies row operations to reach X in echelon form, then applies another set of row operations to reach Y in echelon form.\"},{\"question\":\"What is the key conclusion about echelon-form matrices that are row equivalent to the same matrix?\",\"answer\":\"Multiple echelon-form matrices can be row equivalent to a given matrix, so uniqueness does not hold.\"}]","Linear Equations in Linear Algebra - Chapter 1 - Counterexample on Unique Echelon Form | PDF"]