{"id":2661,"date":"2025-08-13T18:24:07","date_gmt":"2025-08-13T18:24:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2661"},"modified":"2025-09-19T15:55:44","modified_gmt":"2025-09-19T15:55:44","slug":"matrices-and-matrix-operations-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/matrices-and-matrix-operations-background-youll-need-1\/","title":{"raw":"Matrices and Matrix Operations: Background You'll Need 1","rendered":"Matrices and Matrix Operations: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use the commutative and associative properties of numbers to solve math problems.<\/li>\r\n<\/ul>\r\n<\/section>For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.\r\n<h2>Commutative Properties<\/h2>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.\r\n<div style=\"text-align: center;\">[latex]a+b=b+a[\/latex]<\/div>\r\n<div>\r\n\r\nThe <strong>commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.\r\n<div style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Examples:\r\n<ul>\r\n \t<li>[latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/li>\r\n \t<li>[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/li>\r\n<\/ul>\r\nIt is important to note that neither subtraction nor division is commutative.\r\n\r\nNon-examples:\r\n<ul>\r\n \t<li>[latex]17 - 5[\/latex] is not the same as [latex]5 - 17[\/latex].<\/li>\r\n \t<li>[latex]20\\div 5\\ne 5\\div 20[\/latex].<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24864[\/ohm2_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24865[\/ohm2_question]<\/section>\r\n<h2>Associative Properties<\/h2>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>associative property of multiplication<\/strong> tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.\r\n<div style=\"text-align: center;\">[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/div>\r\n<div>\r\n\r\nThe <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.\r\n<div style=\"text-align: center;\">[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/div>\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Examples:\r\n<ul>\r\n \t<li>[latex]\\left(3\\cdot4\\right)\\cdot5=60\\text{ and }3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/li>\r\n \t<li>[latex][15+\\left(-9\\right)]+23=29\\text{ and }15+[\\left(-9\\right)+23]=29[\/latex]<\/li>\r\n<\/ul>\r\nNon-examples:\r\n<ul>\r\n \t<li>\r\n<div style=\"text-align: center;\">[latex]\\begin{align}8-\\left(3-15\\right) &amp; \\stackrel{?}{=}\\left(8-3\\right)-15 \\\\ 8-\\left(-12\\right) &amp; \\stackrel{?}=5-15 \\\\ 20 &amp; \\neq 20-10 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\n<div style=\"text-align: center;\"><\/div><\/li>\r\n \t<li>\r\n<div style=\"text-align: center;\">[latex]\\begin{align}64\\div\\left(8\\div4\\right)&amp;\\stackrel{?}{=}\\left(64\\div8\\right)\\div4 \\\\ 64\\div2 &amp; \\stackrel{?}{=}8\\div4 \\\\ 32 &amp; \\neq 2 \\\\ \\text{ }\\end{align}[\/latex]<\/div><\/li>\r\n<\/ul>\r\nNote: neither subtraction nor division is associative.\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24866[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use the commutative and associative properties of numbers to solve math problems.<\/li>\n<\/ul>\n<\/section>\n<p>For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.<\/p>\n<h2>Commutative Properties<\/h2>\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.<\/p>\n<div style=\"text-align: center;\">[latex]a+b=b+a[\/latex]<\/div>\n<div>\n<p>The <strong>commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Examples:<\/p>\n<ul>\n<li>[latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/li>\n<li>[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/li>\n<\/ul>\n<p>It is important to note that neither subtraction nor division is commutative.<\/p>\n<p>Non-examples:<\/p>\n<ul>\n<li>[latex]17 - 5[\/latex] is not the same as [latex]5 - 17[\/latex].<\/li>\n<li>[latex]20\\div 5\\ne 5\\div 20[\/latex].<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24864\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24864&theme=lumen&iframe_resize_id=ohm24864&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24865\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24865&theme=lumen&iframe_resize_id=ohm24865&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Associative Properties<\/h2>\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>associative property of multiplication<\/strong> tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.<\/p>\n<div style=\"text-align: center;\">[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/div>\n<div>\n<p>The <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.<\/p>\n<div style=\"text-align: center;\">[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Examples:<\/p>\n<ul>\n<li>[latex]\\left(3\\cdot4\\right)\\cdot5=60\\text{ and }3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/li>\n<li>[latex][15+\\left(-9\\right)]+23=29\\text{ and }15+[\\left(-9\\right)+23]=29[\/latex]<\/li>\n<\/ul>\n<p>Non-examples:<\/p>\n<ul>\n<li>\n<div style=\"text-align: center;\">[latex]\\begin{align}8-\\left(3-15\\right) & \\stackrel{?}{=}\\left(8-3\\right)-15 \\\\ 8-\\left(-12\\right) & \\stackrel{?}=5-15 \\\\ 20 & \\neq 20-10 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<div style=\"text-align: center;\"><\/div>\n<\/li>\n<li>\n<div style=\"text-align: center;\">[latex]\\begin{align}64\\div\\left(8\\div4\\right)&\\stackrel{?}{=}\\left(64\\div8\\right)\\div4 \\\\ 64\\div2 & \\stackrel{?}{=}8\\div4 \\\\ 32 & \\neq 2 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ul>\n<p>Note: neither subtraction nor division is associative.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24866\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24866&theme=lumen&iframe_resize_id=ohm24866&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":514,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2661"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":2,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2661\/revisions"}],"predecessor-version":[{"id":4187,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2661\/revisions\/4187"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/514"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2661\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2661"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2661"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2661"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2661"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}