{"id":7901,"date":"2022-05-16T15:23:23","date_gmt":"2022-05-16T09:53:23","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7901"},"modified":"2025-06-02T20:45:38","modified_gmt":"2025-06-02T15:15:38","slug":"what-are-matrices-and-their-types","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/","title":{"rendered":"What are matrices and their types?"},"content":{"rendered":"\n
Matrices have a wide range of applications in various disciplines such as chemistry, Biology, Engineering, Statistics, economics, etc. Matrices play an important role in computer science also. We widely use matrices to solve the system of linear equations, system of linear differential equations and nonlinear differential equations. First time the matrices were introduced by Cayley in 1860.<\/p>\n\n\n\n
Definition<\/strong><\/h2>\n\n\n\n
A matrix is a rectangular arrangement of the numbers. We call these numbers inside the matrix as elements of the matrix.<\/p>\n\n\n\n
HereThe expression of a matrix \u2018A\u2019 is-<\/p>\n\n\n\n<\/figure>\n\n\n\n
The vertical elements are columns and the horizontal elements are rows of the matrix.<\/p>\n\n\n\n
The order of matrix A is m by n or (m\u00d7 n)<\/p>\n\n\n\n
Notation of a matrix<\/strong><\/h2>\n\n\n\n
We denote a matrix \u2018A\u2019 is as-<\/p>\n\n\n\n<\/figure>\n\n\n\n
Where, i = 1, 2, \u2026\u2026.,m and j = 1,2,3,\u2026\u2026.n<\/p>\n\n\n\n
Here \u2018i\u2019 denotes row and \u2018j\u2019 denotes column.<\/p>\n\n\n\n
Types of matrices<\/strong><\/h2>\n\n\n\n
1. Rectangular matrix-<\/strong><\/p>\n\n\n\n
A matrix in which the number of rows is not equal to the number of columns, then we say that the given matrix is a rectangular matrix.<\/p>\n\n\n\n
Example:<\/p>\n\n\n\n<\/figure>\n\n\n\n
The order of matrix A is 2\u00d73, hence, it has two rows and three columns.<\/p>\n\n\n\n
Matrix A is a rectangular matrix.<\/p>\n\n\n\n
2. Square matrix-<\/strong><\/p>\n\n\n\n
A matrix with equal number of rows and columns, is a square matrix.<\/p>\n\n\n\n
Example:<\/p>\n\n\n\n<\/figure>\n\n\n\n
The order of matrix A is 3 \u00d73 , that means it has three rows and three columns.<\/p>\n\n\n\n
Matrix A is a square matrix.<\/p>\n\n\n\n
3. Row matrix-<\/strong><\/p>\n\n\n\n
A matrix with a single row and any number of columns is called row matrix.<\/p>\n\n\n\n
Example:<\/p>\n\n\n\n<\/figure>\n\n\n\n
4. Column matrix-<\/strong><\/p>\n\n\n\n
A matrix with a single column and any number of rows is called row matrix.<\/p>\n\n\n\n
If all the elements above the leading diagonal of a square matrix are zero, hence, we call it a lower triangular matrix.<\/p>\n\n\n\n
Example –<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"
Matrices have a wide range of applications in various disciplines such as chemistry, Biology, Engineering, Statistics, economics, etc.<\/p>\n","protected":false},"author":3,"featured_media":7669,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31],"tags":[],"class_list":["post-7901","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths"],"yoast_head":"\n
What are matrices and their types? - Goseeko blog<\/title>\n<meta name=\"description\" content=\"Matrices have a wide range of applications in various disciplines such as chemistry, Biology, Engineering, Statistics, economics, etc.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What are matrices and their types? - Goseeko blog\" \/>\n<meta property=\"og:description\" content=\"Matrices have a wide range of applications in various disciplines such as chemistry, Biology, Engineering, Statistics, economics, etc.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/\" \/>\n<meta property=\"og:site_name\" content=\"Goseeko blog\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/goseeko\" \/>\n<meta property=\"article:published_time\" content=\"2022-05-16T09:53:23+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-06-02T15:15:38+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.goseeko.com\/blog\/wp-content\/uploads\/2022\/04\/2209021.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"2000\" \/>\n\t<meta property=\"og:image:height\" content=\"2000\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Team Goseeko\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@goseeko\" \/>\n<meta name=\"twitter:site\" content=\"@goseeko\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Team Goseeko\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/\"},\"author\":{\"name\":\"Team Goseeko\",\"@id\":\"https:\/\/www.goseeko.com\/blog\/#\/schema\/person\/7ec300cd01b6116501943af14d546bae\"},\"headline\":\"What are matrices and their types?\",\"datePublished\":\"2022-05-16T09:53:23+00:00\",\"dateModified\":\"2025-06-02T15:15:38+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/\"},\"wordCount\":464,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/www.goseeko.com\/blog\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2022\/04\/2209021.jpg?fit=2000%2C2000&ssl=1\",\"articleSection\":[\"Maths\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/\",\"url\":\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/\",\"name\":\"What are matrices and their types? - Goseeko blog\",\"isPartOf\":{\"@id\":\"https:\/\/www.goseeko.com\/blog\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2022\/04\/2209021.jpg?fit=2000%2C2000&ssl=1\",\"datePublished\":\"2022-05-16T09:53:23+00:00\",\"dateModified\":\"2025-06-02T15:15:38+00:00\",\"description\":\"Matrices have a wide range of applications in various disciplines such as chemistry, Biology, Engineering, Statistics, economics, etc.\",\"breadcrumb\":{\"@id\":\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#primaryimage\",\"url\":\"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2022\/04\/2209021.jpg?fit=2000%2C2000&ssl=1\",\"contentUrl\":\"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2022\/04\/2209021.jpg?fit=2000%2C2000&ssl=1\",\"width\":2000,\"height\":2000},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/www.goseeko.com\/blog\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"What are matrices and their types?\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.goseeko.com\/blog\/#website\",\"url\":\"https:\/\/www.goseeko.com\/blog\/\",\"name\":\"Goseeko blog\",\"description\":\"Learning beyond college, Students platform for life skills.\",\"publisher\":{\"@id\":\"https:\/\/www.goseeko.com\/blog\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.goseeko.com\/blog\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/www.goseeko.com\/blog\/#organization\",\"name\":\"Goseeko.com\",\"url\":\"https:\/\/www.goseeko.com\/blog\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/www.goseeko.com\/blog\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2021\/09\/GoSeeko_Stacked-logo-01.png?fit=2471%2C2471&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2021\/09\/GoSeeko_Stacked-logo-01.png?fit=2471%2C2471&ssl=1\",\"width\":2471,\"height\":2471,\"caption\":\"Goseeko.com\"},\"image\":{\"@id\":\"https:\/\/www.goseeko.com\/blog\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/goseeko\",\"https:\/\/x.com\/goseeko\",\"https:\/\/www.instagram.com\/goseeko\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/www.goseeko.com\/blog\/#\/schema\/person\/7ec300cd01b6116501943af14d546bae\",\"name\":\"Team Goseeko\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/www.goseeko.com\/blog\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/?s=96&d=mm&r=g\",\"caption\":\"Team Goseeko\"},\"url\":\"https:\/\/www.goseeko.com\/blog\/author\/team-goseeko\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"What are matrices and their types? - Goseeko blog","description":"Matrices have a wide range of applications in various disciplines such as chemistry, Biology, Engineering, Statistics, economics, etc.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/","og_locale":"en_US","og_type":"article","og_title":"What are matrices and their types? - Goseeko blog","og_description":"Matrices have a wide range of applications in various disciplines such as chemistry, Biology, Engineering, Statistics, economics, etc.","og_url":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/","og_site_name":"Goseeko blog","article_publisher":"https:\/\/www.facebook.com\/goseeko","article_published_time":"2022-05-16T09:53:23+00:00","article_modified_time":"2025-06-02T15:15:38+00:00","og_image":[{"width":2000,"height":2000,"url":"https:\/\/www.goseeko.com\/blog\/wp-content\/uploads\/2022\/04\/2209021.jpg","type":"image\/jpeg"}],"author":"Team Goseeko","twitter_card":"summary_large_image","twitter_creator":"@goseeko","twitter_site":"@goseeko","twitter_misc":{"Written by":"Team Goseeko","Est. reading time":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#article","isPartOf":{"@id":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/"},"author":{"name":"Team Goseeko","@id":"https:\/\/www.goseeko.com\/blog\/#\/schema\/person\/7ec300cd01b6116501943af14d546bae"},"headline":"What are matrices and their types?","datePublished":"2022-05-16T09:53:23+00:00","dateModified":"2025-06-02T15:15:38+00:00","mainEntityOfPage":{"@id":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/"},"wordCount":464,"commentCount":0,"publisher":{"@id":"https:\/\/www.goseeko.com\/blog\/#organization"},"image":{"@id":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#primaryimage"},"thumbnailUrl":"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2022\/04\/2209021.jpg?fit=2000%2C2000&ssl=1","articleSection":["Maths"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/","url":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/","name":"What are matrices and their types? - Goseeko blog","isPartOf":{"@id":"https:\/\/www.goseeko.com\/blog\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#primaryimage"},"image":{"@id":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#primaryimage"},"thumbnailUrl":"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2022\/04\/2209021.jpg?fit=2000%2C2000&ssl=1","datePublished":"2022-05-16T09:53:23+00:00","dateModified":"2025-06-02T15:15:38+00:00","description":"Matrices have a wide range of applications in various disciplines such as chemistry, Biology, Engineering, Statistics, economics, etc.","breadcrumb":{"@id":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#primaryimage","url":"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2022\/04\/2209021.jpg?fit=2000%2C2000&ssl=1","contentUrl":"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2022\/04\/2209021.jpg?fit=2000%2C2000&ssl=1","width":2000,"height":2000},{"@type":"BreadcrumbList","@id":"https:\/\/www.goseeko.com\/blog\/what-are-matrices-and-their-types\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.goseeko.com\/blog\/"},{"@type":"ListItem","position":2,"name":"What are matrices and their types?"}]},{"@type":"WebSite","@id":"https:\/\/www.goseeko.com\/blog\/#website","url":"https:\/\/www.goseeko.com\/blog\/","name":"Goseeko blog","description":"Learning beyond college, Students platform for life skills.","publisher":{"@id":"https:\/\/www.goseeko.com\/blog\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.goseeko.com\/blog\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/www.goseeko.com\/blog\/#organization","name":"Goseeko.com","url":"https:\/\/www.goseeko.com\/blog\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.goseeko.com\/blog\/#\/schema\/logo\/image\/","url":"https:\/\/i1.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2021\/09\/GoSeeko_Stacked-logo-01.png?fit=2471%2C2471&ssl=1","contentUrl":"https:\/\/i1.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2021\/09\/GoSeeko_Stacked-logo-01.png?fit=2471%2C2471&ssl=1","width":2471,"height":2471,"caption":"Goseeko.com"},"image":{"@id":"https:\/\/www.goseeko.com\/blog\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/goseeko","https:\/\/x.com\/goseeko","https:\/\/www.instagram.com\/goseeko\/"]},{"@type":"Person","@id":"https:\/\/www.goseeko.com\/blog\/#\/schema\/person\/7ec300cd01b6116501943af14d546bae","name":"Team Goseeko","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.goseeko.com\/blog\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/?s=96&d=mm&r=g","caption":"Team Goseeko"},"url":"https:\/\/www.goseeko.com\/blog\/author\/team-goseeko\/"}]}},"jetpack_featured_media_url":"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2022\/04\/2209021.jpg?fit=2000%2C2000&ssl=1","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.goseeko.com\/blog\/wp-json\/wp\/v2\/posts\/7901","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.goseeko.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.goseeko.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.goseeko.com\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.goseeko.com\/blog\/wp-json\/wp\/v2\/comments?post=7901"}],"version-history":[{"count":5,"href":"https:\/\/www.goseeko.com\/blog\/wp-json\/wp\/v2\/posts\/7901\/revisions"}],"predecessor-version":[{"id":7936,"href":"https:\/\/www.goseeko.com\/blog\/wp-json\/wp\/v2\/posts\/7901\/revisions\/7936"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.goseeko.com\/blog\/wp-json\/wp\/v2\/media\/7669"}],"wp:attachment":[{"href":"https:\/\/www.goseeko.com\/blog\/wp-json\/wp\/v2\/media?parent=7901"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.goseeko.com\/blog\/wp-json\/wp\/v2\/categories?post=7901"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.goseeko.com\/blog\/wp-json\/wp\/v2\/tags?post=7901"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
![\"\"](\"https:\/\/lh5.googleusercontent.com\/GY_0bdMJ54CzAVBc9_Xh22V7UKp5RavxZrIvBWEqMj6iVt4G0HXuSUI6ru3dj51Y0tyjj5VJPXnKaZH31JM9RHKQ-3CY9LS-YzMzm5zsYMfndZpdJT7JmQE6CYouoVLX1J8xoxd2WYIsFEbNpQ\")
![\"\"](\"https:\/\/lh5.googleusercontent.com\/-cInNaPikCRrKen7dZnbXYY20EQJPNtA5R04DX1f16TCjWtrKiMQiGTcZNVT0lg9cUFe9wfgnL1-RYpsGT_k6djPcH_LMQwqMDZ5Na-BicaDfLIqAqavIZKLp0CjsHm1NWUujr3MdYgbq7ow_Q\")
![\"\"](\"https:\/\/lh4.googleusercontent.com\/heUqDhIru0QeNlZcQ5HX9WyEnubezbAorVeebdoPXo_61QvhUtRcC3taRlSl5h6EGpSawE_pPpx3BjaSjU3owhBA8zZYPCqAW91NQWy6s6Z7M9VAKaJ07j4TS4ZEhPebYoABCOufDIYOKPrJbQ\")
![\"\"](\"https:\/\/lh4.googleusercontent.com\/2hNVx6kz7NsNg9EVXbMLVUaIihI1dyx9c3GRADFojtv-MnDmzfwaNbbjxzGgA9mA_8m54IFyAUUaLei4QSHxZG0ALhVBs5vfi68f6awO8sI5OgcAzas3MVvVki1RbQ61aFNQKcldV5JMN6JsOw\")
![\"\"](\"https:\/\/lh3.googleusercontent.com\/TEqcWuyC3MiNOnRJGp-w2jsG55BO3WWYQFqqGdw8t9dkhEh0HN3nvYLHBbpquchaNJbOOBN0QCnns-UCdjIvp3oqnklA2s5bA39hwTKWfJQzVE3zovcuhlN6dshtoldy8UevWATr-9uUwYtyaA\")
![\"\"](\"https:\/\/lh4.googleusercontent.com\/xIrubEXiyNRJiC2aV-WK7q47QpupxgVs3PCIwrF4tkRb2WXcsxX4YmX4FsMbr-2W085jt-c2JGxuO7DHc0V6uyDaYuod7W_EJSKVQB_QDKh16u91n3SFxStPw6SGzSDHWr1NoqfiC0UYAF_EKg\")
![\"\"](\"https:\/\/lh3.googleusercontent.com\/qQeTnRoePCTWoK1ndGb86FLHbnE4WvgmAuMXTTTLXOwvUvValuHm9ZXxY0mTB351VoLe2EuwMNYlRAo8R7UCbZgK61NkQnU7I0-GED8rDIIoD80Zi7NSvh_tRx3lE2kcAKiVq189abRW4HD2zQ\")
![](\"https:\/\/lh6.googleusercontent.com\/qM_paaRAbpk8LukOnN0j1Apxn1YmgZEGEyoAGnge1AIN_Eh0Ruy4doiW3ykbRPgaJFd77zbS8WbZN_tyYhis_8gWOTj7-PeY7X2pLMhWTSyorWYm3scCqVni_EYRMJerB7KvR-OsXtcqmVFjOQ\")
![](\"https:\/\/lh6.googleusercontent.com\/D7_pYRl_Y9cYzLu64xrXGDWhOQm4NNiZW1q3muwI9dChUsTtpcUIa8dFU1yHDIGnBFveYAm62fxS8CBNFDApEOpPDwMMZTV_ewY_xRdEUVbHH3IxOF_F1bwh_RPBjZ9ec4ScM9hyANBduIKNDw\")
![\"\"](\"https:\/\/lh3.googleusercontent.com\/g8K2WlYlWJilRFwdhylej9b3gVY6SdZk8_RkNGyZcDWMdSvrQ46tYcjLRbb9OJDwKC-PruMQQrDoksQMlgjX9OtLyc6RBsUXChZzTT4nmgjzaeqqGO_zXfN1gVbrMt6gsqJ2WdEp0RKO2XbeRw\")
![\"\"](\"https:\/\/lh6.googleusercontent.com\/QzHT4StExuLKf7jBdniwuU9daWFpuW6dcgpMc7mWhLJtbE7VudV8L2Dy7NusPxGucC7r-gci84fB1zrZcm56jfwATpRv2jeRCW8AVEK4hLePpM9oNyHp3-tKbZzgT6mSE8Gv6hdaKJnpV6Lb_g\")
![\"\"](\"https:\/\/lh5.googleusercontent.com\/1RHxQVzH0cxnA222I3OmwuyuTowUS8tEU_Aouojeaamb1SMROmPI8VDGWh8AgUwkT7eyUoZ920ffkOeC-WBffRlEvdXqDnT4oTS4we0zcOwtGiqUkPi4j2rk2GvRa6IW6hBferhnvUKzzelIPg\")
![\"\"](\"https:\/\/lh6.googleusercontent.com\/VRRaVa1h68Gkq8wtzfWIxgpDTzhPxZ0KIUU1aeaJKhit7zfPqdrZJ0G-pzwhT4-yHGgvyk_czQV7kHETBwHXb3moA7lrGlIcQNmaUmgLMOUPPrMvwckIdFxPsL09v_jov_0Z0AkGKuQWqySbqg\")