{"id":3188,"date":"2025-08-15T22:33:07","date_gmt":"2025-08-15T22:33:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3188"},"modified":"2025-09-22T19:50:14","modified_gmt":"2025-09-22T19:50:14","slug":"matrices-and-matrix-operations-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/matrices-and-matrix-operations-apply-it\/","title":{"raw":"Matrices and Matrix Operations: Apply It","rendered":"Matrices and Matrix Operations: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the sum and difference of two matrices.<\/li>\r\n \t<li>Find scalar multiples of a matrix.<\/li>\r\n \t<li>Find the product of two matrices.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<p class=\"whitespace-normal break-words\">A regional health department is analyzing nutrition data from school cafeterias across three districts to evaluate and improve meal programs. They need to organize complex data about calories, nutrients, and costs across different schools and meal types. Matrix operations help them efficiently calculate totals, compare programs, and optimize budgets.<\/p>\r\n<p class=\"whitespace-normal break-words\">The health department collected data on three key nutrients (protein, fiber, and vitamin C) for breakfast and lunch programs across three school districts:<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>District A Nutrition Data (grams per meal):<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Breakfast: 12g protein, 4g fiber, 15mg vitamin C<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Lunch: 25g protein, 5g fiber, 10mg vitamin C<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>District B Nutrition Data (grams per meal):<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Breakfast: 15g protein, 6g fiber, 20mg vitamin C<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Lunch: 22g protein, 4g fiber, 35mg vitamin C<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>District C Nutrition Data (grams per meal):<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Breakfast: 10g protein, 4g fiber, 20mg vitamin C<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Lunch: 28g protein, 4g fiber, 5mg vitamin C<\/li>\r\n<\/ul>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312548[\/ohm_question]<\/section>The department wants to find the average nutritional content when combining all three district programs and then dividing those values by [latex]3[\/latex].\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]312550[\/ohm_question]<\/section>\r\n<p class=\"whitespace-normal break-words\">What if the department also tracks cost?\u00a0 Each gram of protein costs $0.15, each gram of fiber costs $0.08, and each mg of vitamin C costs $0.02.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"123\" data-end=\"386\">We have been organizing each district\u2019s nutrition data as a [latex]2\\times 3[\/latex] matrix with rows = meals (Breakfast, Lunch) and columns = nutrients (Protein, Fiber, Vitamin C). For matrix multiplication to work, the inner dimensions must match.<\/p>\r\n\r\n<ul>\r\n \t<li data-start=\"123\" data-end=\"386\">District matrix: [latex]2\\times 3[\/latex] (meals \u00d7 nutrients.<\/li>\r\n \t<li data-start=\"123\" data-end=\"386\">Price matrix: must be [latex]3\\times 1[\/latex] (nutrients \u00d7 cost per unit.<\/li>\r\n \t<li data-start=\"123\" data-end=\"386\">Product: [latex](2\\times 3)(3\\times 1) = 2\\times 1[\/latex], which returns a cost per meal (Breakfast, Lunch).<\/li>\r\n<\/ul>\r\nFind the cost of each meal for District A by multiplying the nutrition data with the cost matrix.\r\n\r\n[reveal-answer q=\"579806\"]Show Solution[\/reveal-answer][hidden-answer a=\"579806\"]\r\n\r\nNutrition data:\r\n[latex]A= \\begin{bmatrix} 12 &amp; 4 &amp; 15\\\\ 25 &amp; 5 &amp; 10 \\end{bmatrix}[\/latex]\r\n\r\nCost matrix:\r\n[latex]\\begin{bmatrix} 0.15\\\\ 0.08\\\\ 0.02 \\end{bmatrix}[\/latex]\r\n\r\n[latex]\\begin{align} A\\mathbf{p}\r\n&amp;=\r\n\\begin{bmatrix}\r\n12 &amp; 4 &amp; 15\\\\\r\n25 &amp; 5 &amp; 10\r\n\\end{bmatrix}\r\n\\begin{bmatrix}\r\n0.15\\\\\r\n0.08\\\\\r\n0.02\r\n\\end{bmatrix} \\\\\\\\\r\n&amp;=\r\n\\begin{bmatrix}\r\n12(0.15)+4(0.08)+15(0.02)\\\\[2pt]\r\n25(0.15)+5(0.08)+10(0.02)\r\n\\end{bmatrix} \\\\\\\\\r\n&amp;=\r\n\\begin{bmatrix}\r\n1.80+0.32+0.30\\\\[2pt]\r\n3.75+0.40+0.20\r\n\\end{bmatrix}\\\\\\\\\r\n&amp;=\r\n\\begin{bmatrix}\r\n2.42\\\\[2pt]\r\n4.35\r\n\\end{bmatrix}\r\n\\end{align}[\/latex]\r\n\r\nFor District A the breakfast cost is $2.42 per meal and the lunch cost is $4.35 per meal.\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]312551[\/ohm_question]<\/p>\r\n\r\n<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the sum and difference of two matrices.<\/li>\n<li>Find scalar multiples of a matrix.<\/li>\n<li>Find the product of two matrices.<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<p class=\"whitespace-normal break-words\">A regional health department is analyzing nutrition data from school cafeterias across three districts to evaluate and improve meal programs. They need to organize complex data about calories, nutrients, and costs across different schools and meal types. Matrix operations help them efficiently calculate totals, compare programs, and optimize budgets.<\/p>\n<p class=\"whitespace-normal break-words\">The health department collected data on three key nutrients (protein, fiber, and vitamin C) for breakfast and lunch programs across three school districts:<\/p>\n<p class=\"whitespace-normal break-words\"><strong>District A Nutrition Data (grams per meal):<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Breakfast: 12g protein, 4g fiber, 15mg vitamin C<\/li>\n<li class=\"whitespace-normal break-words\">Lunch: 25g protein, 5g fiber, 10mg vitamin C<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>District B Nutrition Data (grams per meal):<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Breakfast: 15g protein, 6g fiber, 20mg vitamin C<\/li>\n<li class=\"whitespace-normal break-words\">Lunch: 22g protein, 4g fiber, 35mg vitamin C<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>District C Nutrition Data (grams per meal):<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Breakfast: 10g protein, 4g fiber, 20mg vitamin C<\/li>\n<li class=\"whitespace-normal break-words\">Lunch: 28g protein, 4g fiber, 5mg vitamin C<\/li>\n<\/ul>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312548\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312548&theme=lumen&iframe_resize_id=ohm312548&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>The department wants to find the average nutritional content when combining all three district programs and then dividing those values by [latex]3[\/latex].<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm312550\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312550&theme=lumen&iframe_resize_id=ohm312550&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p class=\"whitespace-normal break-words\">What if the department also tracks cost?\u00a0 Each gram of protein costs $0.15, each gram of fiber costs $0.08, and each mg of vitamin C costs $0.02.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"123\" data-end=\"386\">We have been organizing each district\u2019s nutrition data as a [latex]2\\times 3[\/latex] matrix with rows = meals (Breakfast, Lunch) and columns = nutrients (Protein, Fiber, Vitamin C). For matrix multiplication to work, the inner dimensions must match.<\/p>\n<ul>\n<li data-start=\"123\" data-end=\"386\">District matrix: [latex]2\\times 3[\/latex] (meals \u00d7 nutrients.<\/li>\n<li data-start=\"123\" data-end=\"386\">Price matrix: must be [latex]3\\times 1[\/latex] (nutrients \u00d7 cost per unit.<\/li>\n<li data-start=\"123\" data-end=\"386\">Product: [latex](2\\times 3)(3\\times 1) = 2\\times 1[\/latex], which returns a cost per meal (Breakfast, Lunch).<\/li>\n<\/ul>\n<p>Find the cost of each meal for District A by multiplying the nutrition data with the cost matrix.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q579806\">Show Solution<\/button><\/p>\n<div id=\"q579806\" class=\"hidden-answer\" style=\"display: none\">\n<p>Nutrition data:<br \/>\n[latex]A= \\begin{bmatrix} 12 & 4 & 15\\\\ 25 & 5 & 10 \\end{bmatrix}[\/latex]<\/p>\n<p>Cost matrix:<br \/>\n[latex]\\begin{bmatrix} 0.15\\\\ 0.08\\\\ 0.02 \\end{bmatrix}[\/latex]<\/p>\n<p>[latex]\\begin{align} A\\mathbf{p}  &=  \\begin{bmatrix}  12 & 4 & 15\\\\  25 & 5 & 10  \\end{bmatrix}  \\begin{bmatrix}  0.15\\\\  0.08\\\\  0.02  \\end{bmatrix} \\\\\\\\  &=  \\begin{bmatrix}  12(0.15)+4(0.08)+15(0.02)\\\\[2pt]  25(0.15)+5(0.08)+10(0.02)  \\end{bmatrix} \\\\\\\\  &=  \\begin{bmatrix}  1.80+0.32+0.30\\\\[2pt]  3.75+0.40+0.20  \\end{bmatrix}\\\\\\\\  &=  \\begin{bmatrix}  2.42\\\\[2pt]  4.35  \\end{bmatrix}  \\end{align}[\/latex]<\/p>\n<p>For District A the breakfast cost is $2.42 per meal and the lunch cost is $4.35 per meal.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm312551\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=312551&theme=lumen&iframe_resize_id=ohm312551&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":67,"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":514,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3188"}],"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":11,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3188\/revisions"}],"predecessor-version":[{"id":4263,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3188\/revisions\/4263"}],"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\/3188\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3188"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3188"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3188"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}