{"id":2455,"date":"2024-07-29T22:38:06","date_gmt":"2024-07-29T22:38:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2455"},"modified":"2024-12-02T16:21:05","modified_gmt":"2024-12-02T16:21:05","slug":"introduction-to-matrices-and-matrix-operations-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-matrices-and-matrix-operations-apply-it-1\/","title":{"raw":"Introduction to Matrices and Matrix Operations: Apply It 1","rendered":"Introduction to Matrices and Matrix Operations: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Understand how to add and subtract matrices<\/li>\r\n \t<li>Figure out how to multiply a matrix by a number (scalar) to change its scale<\/li>\r\n \t<li>Learn to multiply two matrices together to find a new matrix<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Using Matrices in Real-World Problems<\/h2>\r\n<section class=\"textbox example\" aria-label=\"Example\">Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment from the fall 2013 semester to the fall of 2014. They estimate that [latex]15\\%[\/latex] more equipment is needed in both labs. The school\u2019s current inventory is displayed in the table below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">Lab [latex]A[\/latex]<\/th>\r\n<th style=\"text-align: center;\">Lab [latex]B[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><strong>Computers<\/strong><\/td>\r\n<td style=\"text-align: center;\">[latex]15[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]27[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Computer Tables<\/strong><\/td>\r\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]34[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Chairs<\/strong><\/td>\r\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]34[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nConvert the data to a matrix and find the inventory for 2014.\r\n\r\nComputer inventory in fall 2013:\r\n<p style=\"text-align: center;\">[latex]{C}_{2013}=\\left[\\begin{array}{c}15\\\\ 16\\\\ 16\\end{array}\\begin{array}{c}27\\\\ 34\\\\ 34\\end{array}\\right][\/latex]<\/p>\r\nTo calculate how much computer equipment will be needed in 2014, we multiply all entries in matrix [latex]C[\/latex] by [latex]0.15[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left(0.15\\right){C}_{2013}=\\left[\\begin{array}{c}\\left(0.15\\right)15\\\\ \\left(0.15\\right)16\\\\ \\left(0.15\\right)16\\end{array}\\begin{array}{c}\\left(0.15\\right)27\\\\ \\left(0.15\\right)34\\\\ \\left(0.15\\right)34\\end{array}\\right]=\\left[\\begin{array}{c}2.25\\\\ 2.4\\\\ 2.4\\end{array}\\begin{array}{c}4.05\\\\ 5.1\\\\ 5.1\\end{array}\\right][\/latex]<\/p>\r\nWe must round up to the next integer, so the amount of new equipment needed is\r\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{c}3\\\\ 3\\\\ 3\\end{array}\\begin{array}{c}5\\\\ 6\\\\ 6\\end{array}\\right][\/latex]<\/p>\r\nAdding the two matrices as shown below, we see the new inventory amounts.\r\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{c}15\\\\ 16\\\\ 16\\end{array}\\begin{array}{c}27\\\\ 34\\\\ 34\\end{array}\\right]+\\left[\\begin{array}{c}3\\\\ 3\\\\ 3\\end{array}\\begin{array}{c}5\\\\ 6\\\\ 6\\end{array}\\right]=\\left[\\begin{array}{c}18\\\\ 19\\\\ 19\\end{array}\\begin{array}{c}32\\\\ 40\\\\ 40\\end{array}\\right][\/latex]<\/p>\r\nThis means\r\n<p style=\"text-align: center;\">[latex]{C}_{2014}=\\left[\\begin{array}{c}18\\\\ 19\\\\ 19\\end{array}\\begin{array}{c}32\\\\ 40\\\\ 40\\end{array}\\right][\/latex]<\/p>\r\nThus, Lab A will have [latex]18[\/latex] computers, [latex]19[\/latex] computer tables, and [latex]19[\/latex] chairs; Lab B will have [latex]32[\/latex] computers, [latex]40[\/latex] computer tables, and [latex]40[\/latex] chairs.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season.The table below, representing the equipment needs of two soccer teams.\r\n<table summary=\"..\">\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">Wildcats<\/th>\r\n<th style=\"text-align: center;\">Mud Cats<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><strong>Goals<\/strong><\/td>\r\n<td style=\"text-align: center;\">[latex]6[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Balls<\/strong><\/td>\r\n<td style=\"text-align: center;\">[latex]30[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Jerseys<\/strong><\/td>\r\n<td style=\"text-align: center;\">[latex]14[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]20[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe are also given the prices of the equipment, as shown in the table below.\r\n<table summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Goal<\/strong><\/td>\r\n<td>[latex]$300[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Ball<\/strong><\/td>\r\n<td>[latex]$10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Jersey<\/strong><\/td>\r\n<td>[latex]$30[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nHow can we find the total cost for the equipment needed for each team?\r\n\r\n[reveal-answer q=\"504187\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"504187\"]\r\n\r\nWe will convert the data to matrices. Thus, the equipment need matrix is written as\r\n<p style=\"text-align: center;\">[latex]E=\\left[\\begin{array}{c}6\\\\ 30\\\\ 14\\end{array}\\begin{array}{c}10\\\\ 24\\\\ 20\\end{array}\\right][\/latex]<\/p>\r\nThe cost matrix is written as\r\n<p style=\"text-align: center;\">[latex]C=\\left[\\begin{array}{ccc}300&amp; 10&amp; 30\\end{array}\\right][\/latex]\r\nWe perform matrix multiplication to obtain costs for the equipment.\r\n[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ CE &amp; =\\left[\\begin{array}{rrr}\\hfill 300&amp; \\hfill 10&amp; \\hfill 30\\end{array}\\right]\\cdot \\left[\\begin{array}{rr}\\hfill 6&amp; \\hfill 10\\\\ \\hfill 30&amp; \\hfill 24\\\\ \\hfill 14&amp; \\hfill 20\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rr}\\hfill 300\\left(6\\right)+10\\left(30\\right)+30\\left(14\\right)&amp; \\hfill 300\\left(10\\right)+10\\left(24\\right)+30\\left(20\\right)\\end{array}\\right]\\hfill \\\\ &amp; =\\left[\\begin{array}{rr}\\hfill 2,520&amp; \\hfill 3,840\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\r\nThe total cost for equipment for the Wildcats is [latex]$2,520[\/latex], and the total cost for equipment for the Mud Cats is [latex]$3,840[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]293729[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Understand how to add and subtract matrices<\/li>\n<li>Figure out how to multiply a matrix by a number (scalar) to change its scale<\/li>\n<li>Learn to multiply two matrices together to find a new matrix<\/li>\n<\/ul>\n<\/section>\n<h2>Using Matrices in Real-World Problems<\/h2>\n<section class=\"textbox example\" aria-label=\"Example\">Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment from the fall 2013 semester to the fall of 2014. They estimate that [latex]15\\%[\/latex] more equipment is needed in both labs. The school\u2019s current inventory is displayed in the table below.<\/p>\n<table>\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">Lab [latex]A[\/latex]<\/th>\n<th style=\"text-align: center;\">Lab [latex]B[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Computers<\/strong><\/td>\n<td style=\"text-align: center;\">[latex]15[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]27[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Computer Tables<\/strong><\/td>\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]34[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Chairs<\/strong><\/td>\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]34[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Convert the data to a matrix and find the inventory for 2014.<\/p>\n<p>Computer inventory in fall 2013:<\/p>\n<p style=\"text-align: center;\">[latex]{C}_{2013}=\\left[\\begin{array}{c}15\\\\ 16\\\\ 16\\end{array}\\begin{array}{c}27\\\\ 34\\\\ 34\\end{array}\\right][\/latex]<\/p>\n<p>To calculate how much computer equipment will be needed in 2014, we multiply all entries in matrix [latex]C[\/latex] by [latex]0.15[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left(0.15\\right){C}_{2013}=\\left[\\begin{array}{c}\\left(0.15\\right)15\\\\ \\left(0.15\\right)16\\\\ \\left(0.15\\right)16\\end{array}\\begin{array}{c}\\left(0.15\\right)27\\\\ \\left(0.15\\right)34\\\\ \\left(0.15\\right)34\\end{array}\\right]=\\left[\\begin{array}{c}2.25\\\\ 2.4\\\\ 2.4\\end{array}\\begin{array}{c}4.05\\\\ 5.1\\\\ 5.1\\end{array}\\right][\/latex]<\/p>\n<p>We must round up to the next integer, so the amount of new equipment needed is<\/p>\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{c}3\\\\ 3\\\\ 3\\end{array}\\begin{array}{c}5\\\\ 6\\\\ 6\\end{array}\\right][\/latex]<\/p>\n<p>Adding the two matrices as shown below, we see the new inventory amounts.<\/p>\n<p style=\"text-align: center;\">[latex]\\left[\\begin{array}{c}15\\\\ 16\\\\ 16\\end{array}\\begin{array}{c}27\\\\ 34\\\\ 34\\end{array}\\right]+\\left[\\begin{array}{c}3\\\\ 3\\\\ 3\\end{array}\\begin{array}{c}5\\\\ 6\\\\ 6\\end{array}\\right]=\\left[\\begin{array}{c}18\\\\ 19\\\\ 19\\end{array}\\begin{array}{c}32\\\\ 40\\\\ 40\\end{array}\\right][\/latex]<\/p>\n<p>This means<\/p>\n<p style=\"text-align: center;\">[latex]{C}_{2014}=\\left[\\begin{array}{c}18\\\\ 19\\\\ 19\\end{array}\\begin{array}{c}32\\\\ 40\\\\ 40\\end{array}\\right][\/latex]<\/p>\n<p>Thus, Lab A will have [latex]18[\/latex] computers, [latex]19[\/latex] computer tables, and [latex]19[\/latex] chairs; Lab B will have [latex]32[\/latex] computers, [latex]40[\/latex] computer tables, and [latex]40[\/latex] chairs.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season.The table below, representing the equipment needs of two soccer teams.<\/p>\n<table summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">Wildcats<\/th>\n<th style=\"text-align: center;\">Mud Cats<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Goals<\/strong><\/td>\n<td style=\"text-align: center;\">[latex]6[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Balls<\/strong><\/td>\n<td style=\"text-align: center;\">[latex]30[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Jerseys<\/strong><\/td>\n<td style=\"text-align: center;\">[latex]14[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]20[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We are also given the prices of the equipment, as shown in the table below.<\/p>\n<table summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Goal<\/strong><\/td>\n<td>[latex]$300[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Ball<\/strong><\/td>\n<td>[latex]$10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Jersey<\/strong><\/td>\n<td>[latex]$30[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>How can we find the total cost for the equipment needed for each team?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q504187\">Show Answer<\/button><\/p>\n<div id=\"q504187\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will convert the data to matrices. Thus, the equipment need matrix is written as<\/p>\n<p style=\"text-align: center;\">[latex]E=\\left[\\begin{array}{c}6\\\\ 30\\\\ 14\\end{array}\\begin{array}{c}10\\\\ 24\\\\ 20\\end{array}\\right][\/latex]<\/p>\n<p>The cost matrix is written as<\/p>\n<p style=\"text-align: center;\">[latex]C=\\left[\\begin{array}{ccc}300& 10& 30\\end{array}\\right][\/latex]<br \/>\nWe perform matrix multiplication to obtain costs for the equipment.<br \/>\n[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ CE & =\\left[\\begin{array}{rrr}\\hfill 300& \\hfill 10& \\hfill 30\\end{array}\\right]\\cdot \\left[\\begin{array}{rr}\\hfill 6& \\hfill 10\\\\ \\hfill 30& \\hfill 24\\\\ \\hfill 14& \\hfill 20\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rr}\\hfill 300\\left(6\\right)+10\\left(30\\right)+30\\left(14\\right)& \\hfill 300\\left(10\\right)+10\\left(24\\right)+30\\left(20\\right)\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rr}\\hfill 2,520& \\hfill 3,840\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/p>\n<p>The total cost for equipment for the Wildcats is [latex]$2,520[\/latex], and the total cost for equipment for the Mud Cats is [latex]$3,840[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm293729\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=293729&theme=lumen&iframe_resize_id=ohm293729&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":12,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":327,"module-header":"apply_it","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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2455"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2455\/revisions"}],"predecessor-version":[{"id":6529,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2455\/revisions\/6529"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/327"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2455\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2455"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2455"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2455"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2455"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}