Receiving Helpdesk

4 by 4 matrix

by Dr. Terrence Nikolaus Published 4 years ago Updated 2 years ago

Determinant of a 4×4 matrix is a unique number which is calculated using a particular formula. If a matrix order is n x n, then it is a square matrix. Hence, here 4×4 is a square matrix which has four rows and four columns. If A is square matrix then the determinant of matrix A is represented as |A|.

Full Answer

How to multiply 4x4 matrix?

Properties of Matrix Multiplication

  • Commutative Property. The matrix multiplication is not commutative. ...
  • Associative Property. Hence, the associative property of matrix multiplication is proved.
  • Distributive Property
  • Multiplicative Identity Property. Where A is an n× n matrix and “I” is an identity matrix of order n. ...
  • Dimension Property. ...
  • Multiplicative property of Zero. ...

What is 4 multiplied by 4?

Multiply by 4 and learn the 4 times table with a tropical beat. Sing along to the 4's times tables by singing along with the multiplication fact the first...

How to solve matrix 4x4?

Determinant of 4x4 Matrix. Determinant of a 4×4 matrix is a unique number which is calculated using a particular formula. If a matrix order is n x n, then it is a square matrix. Hence, here 4×4 is a square matrix which has four rows and four columns. If A is square matrix then the determinant of matrix A is represented as |A|.

How to find the inverse matrix of a 4x4 matrix?

Where:

  • A-1 is the inverse of matrix A
  • det (A) is the determinant of the given matrix
  • adj (A) is the adjoint of the given matrix

How do you solve a 4 by 4 matrix?

4:4011:29How To Find The Determinant of a 4x4 Matrix - YouTubeYouTubeStart of suggested clipEnd of suggested clipAnd then plus the last number in the first row which is three. And that's in row one column threeMoreAnd then plus the last number in the first row which is three. And that's in row one column three which leaves behind two three one negative two.

What is the minor of a 4x4 matrix?

For a 4x4 matrix A, there are 16 different minors. Each minor is the determinant of the matrix obtained from A by removing one row and one column. Specifically, for 1 = < i,j =< 4, the (i,j) minor of A is the determinant of the matrix obtained from A by removing row i and row j.

What is the shortcut to solve a matrix 4x4?

2:306:03How to Find The Determinant of a 4x4 Matrix (Shortcut Method) - YouTubeYouTubeStart of suggested clipEnd of suggested clipMinus 2 times 0 which doesn't change anything 4 minus 2 times 2 when you do that you get the 0 thereMoreMinus 2 times 0 which doesn't change anything 4 minus 2 times 2 when you do that you get the 0 there as desired. And then you're doing 2 minus 2 times 1 which gives you the 0 as desired.

How do you expand a 4x4 matrix?

0:276:33The Determinant of a 4 by 4 Matrix Using Cofactor Expansion ...YouTubeStart of suggested clipEnd of suggested clipOne next we eliminate row four and column one from the matrix. And multiply by the determinant ofMoreOne next we eliminate row four and column one from the matrix. And multiply by the determinant of the remaining matrix.

How do you find a minor in a 4x4?

1:368:48How to Find the Determinant of a 4x4 Matrix | Matrices | Math Dot ComYouTubeStart of suggested clipEnd of suggested clipSo you need to find the determinant of this 3 into 3 matrix. And that determinant will then will beMoreSo you need to find the determinant of this 3 into 3 matrix. And that determinant will then will be the minor of the element.

How do you find the cofactor of a 4x4 matrix?

0:224:23Determinant of a 4 x 4 Matrix Using Cofactors - YouTubeYouTubeStart of suggested clipEnd of suggested clipWhat we do we pick our favorite row or column. Then we're going to take some over the products inMoreWhat we do we pick our favorite row or column. Then we're going to take some over the products in that row or column of the entries times the corresponding cofactor. That gives us our determining.

How do you find the eigenvalues of a 4x4 matrix?

0:008:30Ex: Find the Eigenvalues of a 4x4 Matrix - YouTubeYouTubeStart of suggested clipEnd of suggested clipHere so we want to find the values of lambda to make this determinant equal to zero to evaluate theMoreHere so we want to find the values of lambda to make this determinant equal to zero to evaluate the determinant. We'll use the cofactor method.

How do you find the determinant of a 4x4 matrix using row operations?

2:184:22Determinant of a 4 x 4 Matrix Using Row Operations - YouTubeYouTubeStart of suggested clipEnd of suggested clipYou're going to multiply it by minus 1 to the row plus column number.MoreYou're going to multiply it by minus 1 to the row plus column number.

How do you expand a row in a matrix?

Expanding to Find the DeterminantPick any row or column in the matrix. It does not matter which row or which column you use, the answer will be the same for any row. ... Multiply every element in that row or column by its cofactor and add. The result is the determinant.

How do you expand a 3x3 determinant?

0:416:16Determinant 3 x 3 Matrix Expansion by Minors - YouTubeYouTubeStart of suggested clipEnd of suggested clipWe're going to take a which is my first entry times e times I minus F times H. Well if you take outMoreWe're going to take a which is my first entry times e times I minus F times H. Well if you take out this column. In this row that a is in we have a 2 by 2 matrix. And this part of the formula.

When can you use cofactor expansion?

Cofactor expansion can be very handy when the matrix has many 0's. Let A=[1a0n−1B] where a is 1×(n−1), B is (n−1)×(n−1), and 0n−1 is an (n−1)-tuple of 0's. Using the formula for expanding along column 1, we obtain just one term since Ai,1=0 for all i≥2. Hence, det(A)=(−1)1+1A1,1det(A(1∣1))=1det(B)=det(B).

Instant

Most international development agencies and non-governmental organizations (NGOs) have some experience planning and evaluating projects using the Logical Framework matrix.

What is the Logical Framework?

The Logical Framework is a tool commonly used in international development to plan and evaluate development projects. It is essentially a 4 x 4 matrix [see figure] that was developed for the US Agency for International Development ( USAID) in 1969.

How do managerial traditions inform development projects?

Our point of departure is that performance management tools are not neutral. They are imbued with all sorts of traditions. We examine the traditions implicated in one such management tool, the Logical Framework, by considering its visual design.

So, what are the implications for development projects?

Access to land and food security, for example, implicates all sorts of local and international economic, climatic, cultural, social, and political factors affecting one another.

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 1 2 3 4 5 6 7 8 9
8.3.21PHP Version441msRequest Duration2MBMemory UsageGET {post}Route
  • warninglog[04:08:45] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\QueryFormatter:...
  • warninglog[04:08:45] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\QueryFormatter:...
  • warninglog[04:08:45] LOG.warning: Callables of the form ["Swift_SmtpTransport", "Swift_Transport_EsmtpTranspor...
  • warninglog[04:08:45] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\SimpleFormatter...
  • warninglog[04:08:45] LOG.warning: Creation of dynamic property Barryvdh\Debugbar\DataFormatter\SimpleFormatter...
  • warninglog[04:08:45] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[04:08:45] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[04:08:45] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[04:08:45] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[04:08:45] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[04:08:45] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[04:08:45] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[04:08:45] LOG.warning: json_decode(): Passing null to parameter #1 ($json) of type string is deprec...
  • warninglog[04:08:45] LOG.warning: mt_rand(): Passing null to parameter #2 ($max) of type int is deprecated in ...
  • Booting (17.82ms)
  • Application (423ms)
  • 1 x Application (95.86%)
    422.91ms
    1 x Booting (4.04%)
    17.82ms
    7 templates were rendered
    • themes.DevBlog.content.post (resources/views/themes/DevBlog/content/post.blade.php)34blade
      Params
      0
      post
      1
      postContent
      2
      author
      3
      updated_at
      4
      bing_rich_snippet_text
      5
      bing_rich_snippet_link
      6
      bing_related_keywords
      7
      google_related_keywords
      8
      bing_news_title
      9
      bing_news_description
      10
      bing_videos
      11
      bing_images
      12
      bing_search_result_title
      13
      bing_search_result_description
      14
      bing_search_result_url
      15
      bing_paa_questions
      16
      bing_paa_answers
      17
      bing_slider_faq_questions
      18
      bing_slider_faq_answers
      19
      bing_pop_faq_questions
      20
      bing_pop_faq_answers
      21
      bing_tab_faq_questions
      22
      bing_tab_faq_answers
      23
      google_faq_questions
      24
      google_faq_answers
      25
      google_rich_snippet
      26
      google_search_result
      27
      indexedArray
      28
      total_images
      29
      total_videos
      30
      settings
      31
      url_current
      32
      menus
      33
      sidebar
    • themes.DevBlog.layouts.master (resources/views/themes/DevBlog/layouts/master.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.head (resources/views/themes/DevBlog/panels/head.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.header (resources/views/themes/DevBlog/panels/header.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.navbar (resources/views/themes/DevBlog/panels/navbar.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.footer (resources/views/themes/DevBlog/panels/footer.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    • themes.DevBlog.panels.scripts (resources/views/themes/DevBlog/panels/scripts.blade.php)41blade
      Params
      0
      __env
      1
      app
      2
      errors
      3
      post
      4
      postContent
      5
      author
      6
      updated_at
      7
      bing_rich_snippet_text
      8
      bing_rich_snippet_link
      9
      bing_related_keywords
      10
      google_related_keywords
      11
      bing_news_title
      12
      bing_news_description
      13
      bing_videos
      14
      bing_images
      15
      bing_search_result_title
      16
      bing_search_result_description
      17
      bing_search_result_url
      18
      bing_paa_questions
      19
      bing_paa_answers
      20
      bing_slider_faq_questions
      21
      bing_slider_faq_answers
      22
      bing_pop_faq_questions
      23
      bing_pop_faq_answers
      24
      bing_tab_faq_questions
      25
      bing_tab_faq_answers
      26
      google_faq_questions
      27
      google_faq_answers
      28
      google_rich_snippet
      29
      google_search_result
      30
      indexedArray
      31
      total_images
      32
      total_videos
      33
      settings
      34
      url_current
      35
      menus
      36
      sidebar
      37
      i
      38
      __currentLoopData
      39
      loop
      40
      item
    uri
    GET {post}
    middleware
    web, checkdate
    as
    post.show
    controller
    App\Http\Controllers\Frontend\json_data\PostController@show
    namespace
    where
    file
    app/Http/Controllers/Frontend/json_data/PostController.php:18-166
    7 statements were executed400ms
    • select * from `posts` where `published_at` <= '2025-06-09 04:08:45' and `slug` = '4-by-4-matrix' and `posts`.`deleted_at` is null limit 1
      3.56ms/app/Providers/RouteServiceProvider.php:54receivinghelpdeskask
      Metadata
      Bindings
      • 0. 2025-06-09 04:08:45
      • 1. 4-by-4-matrix
      Backtrace
      • 15. /app/Providers/RouteServiceProvider.php:54
      • 18. /vendor/laravel/framework/src/Illuminate/Routing/Router.php:842
      • 19. Route binding:39
      • 20. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 21. /vendor/laravel/framework/src/Illuminate/Foundation/Http/Middleware/VerifyCsrfToken.php:78
    • select * from `json_post_contents` where `json_post_contents`.`post_id` = 75896 and `json_post_contents`.`post_id` is not null and `rewrite_id` = 0
      8.21msmiddleware::checkdate:30receivinghelpdeskask
      Metadata
      Bindings
      • 0. 75896
      • 1. 0
      Backtrace
      • 19. middleware::checkdate:30
      • 20. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 21. /vendor/laravel/jetstream/src/Http/Middleware/ShareInertiaData.php:61
      • 22. /vendor/laravel/framework/src/Illuminate/Pipeline/Pipeline.php:167
      • 23. /vendor/laravel/framework/src/Illuminate/Routing/Middleware/SubstituteBindings.php:50
    • select * from `nova_menu_menus` where `slug` = 'header' limit 1
      880μs/vendor/outl1ne/nova-menu-builder/src/helpers.php:32receivinghelpdeskask
      Metadata
      Bindings
      • 0. header
      Backtrace
      • 15. /vendor/outl1ne/nova-menu-builder/src/helpers.php:32
      • 17. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 18. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 19. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
      • 20. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
    • select * from `nova_menu_menu_items` where `nova_menu_menu_items`.`menu_id` = 1 and `nova_menu_menu_items`.`menu_id` is not null and `parent_id` is null order by `parent_id` asc, `order` asc, `name` asc
      550μs/vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35receivinghelpdeskask
      Metadata
      Bindings
      • 0. 1
      Backtrace
      • 19. /vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35
      • 20. /vendor/outl1ne/nova-menu-builder/src/helpers.php:33
      • 22. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 23. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 24. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
    • select * from `nova_menu_menu_items` where `nova_menu_menu_items`.`parent_id` in (1) order by `order` asc
      1.08ms/vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35receivinghelpdeskask
      Metadata
      Backtrace
      • 24. /vendor/outl1ne/nova-menu-builder/src/Models/Menu.php:35
      • 25. /vendor/outl1ne/nova-menu-builder/src/helpers.php:33
      • 27. /vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 28. /vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:45
      • 29. /vendor/laravel/framework/src/Illuminate/Routing/Route.php:261
    • select `id`, `post_title`, `slug` from `posts` where `status` = 'publish' and `posts`.`deleted_at` is null order by RAND() limit 10
      385ms/app/View/Composers/SidebarView.php:22receivinghelpdeskask
      Metadata
      Bindings
      • 0. publish
      Backtrace
      • 14. /app/View/Composers/SidebarView.php:22
      • 15. /app/View/Composers/SidebarView.php:12
      • 16. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:124
      • 17. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:162
      • 20. /vendor/laravel/framework/src/Illuminate/View/Concerns/ManagesEvents.php:177
    • select * from `fake_users` where `fake_users`.`id` = 15724 limit 1
      710μsview::2dd102cf0462e89a4d4d8bc77355d767652bf9aa:15receivinghelpdeskask
      Metadata
      Bindings
      • 0. 15724
      Backtrace
      • 21. view::2dd102cf0462e89a4d4d8bc77355d767652bf9aa:15
      • 23. /vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:108
      • 24. /vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 25. /vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
      • 26. /vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:61
    App\Models\FakeUser
    1
    Outl1ne\MenuBuilder\Models\MenuItem
    1
    Outl1ne\MenuBuilder\Models\Menu
    1
    App\Models\JsonPostContent
    1
    App\Models\Post
    11
        _token
        YIZbHgxfoG3uECSjkQip0kCmTrYX0szySBH9DUEt
        _previous
        array:1 [ "url" => "https://receivinghelpdesk.com/ask/4-by-4-matrix" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /4-by-4-matrix
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:25 [ "cookie" => array:1 [ 0 => "_pk_id.64.7c30=9eb14b80eb440b99.1749422317.; _pk_ses.64.7c30=1; XSRF-TOKEN=eyJpdiI6Ikh1V2lQK2tVMXJGOTZUYVdNaGsvYlE9PSIsInZhbHVlIjoiektZNU1Fb2R1Y1RlN3QxUzNVbkxCbGdpSUFER3VyZk1OMzZVNTJOWFJBM0dnNkF3MEx5Q3BiWkw2Yk5WaytYbHU2eHF0WXBsTkg0a2phcGV6QWpVWFJVUCtxbzRyaitQQlk3M1dCRHRBMW11bURKQm04cTBrd0dudko2bTJWWFoiLCJtYWMiOiI5NzMzMzkwODc0OGYwN2YwNjdjOTliYTI0YTA2MGI2ZTNkNzc5YTQxMmU4MTM0ZjM5ZGRlNzhkNTdlNzdkZTE5IiwidGFnIjoiIn0%3D; askhelpdesk_session=eyJpdiI6IkpNVVV6YkRaZXlJQ2toQ21ESW9Uenc9PSIsInZhbHVlIjoiYVY2eEo0T2tuR1lMQzBleGdtbS9YMlJHakhuU2dqc01pQ3ZBNVVYdGJ6RS9JSDVlNy9wemNSWUt0aGw5K1BENko3N2ZLWGsyYlFoOGViaE1ZZmx1VEpwS3I2Ryt4OG54dzBXQWxCYjFpRE9wSlpKSDFXaVo0V3hyaVhJbytDcWciLCJtYWMiOiIyN2I2YWYxNzBiNTc3NGU5MWE1ZDVkNDM2ZmZjOTA4YjlmMWU4ODk0MThkMmU2YmNhYTIwZGMzZWQ1Yzc1YTMxIiwidGFnIjoiIn0%3D_pk_id.64.7c30=9eb14b80eb440b99.1749422317.; _pk_ses.64.7c30=1; XSRF-TOKEN=eyJpdiI6Ikh1V2lQK2tVMXJGOTZUYVdNaGsvYlE9PSIsInZhbHVlIjoiektZNU1Fb2R1Y1RlN3QxUzNVbkxCb" ] "cf-ipcountry" => array:1 [ 0 => "US" ] "cf-connecting-ip" => array:1 [ 0 => "216.73.216.192" ] "cdn-loop" => array:1 [ 0 => "cloudflare; loops=1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""Chromium";v="130", "HeadlessChrome";v="130", "Not?A_Brand";v="99"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] "accept-encoding" => array:1 [ 0 => "gzip, br" ] "cf-ray" => array:1 [ 0 => "94cbe19f0badf92b-ORD" ] "priority" => array:1 [ 0 => "u=0, i" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "cf-visitor" => array:1 [ 0 => "{"scheme":"https"}" ] "connection" => array:1 [ 0 => "close" ] "x-forwarded-proto" => array:1 [ 0 => "https" ] "x-forwarded-for" => array:1 [ 0 => "216.73.216.192, 172.69.7.208" ] "x-server-addr" => array:1 [ 0 => "154.12.239.204" ] "host" => array:1 [ 0 => "receivinghelpdesk.com" ] ]
        request_server
        0 of 0
        array:56 [ "USER" => "runcloud" "HOME" => "/home/runcloud" "SCRIPT_NAME" => "/ask/index.php" "REQUEST_URI" => "/ask/4-by-4-matrix" "QUERY_STRING" => "" "REQUEST_METHOD" => "GET" "SERVER_PROTOCOL" => "HTTP/1.0" "GATEWAY_INTERFACE" => "CGI/1.1" "REDIRECT_URL" => "/ask/4-by-4-matrix" "REMOTE_PORT" => "54312" "SCRIPT_FILENAME" => "/home/runcloud/webapps/ReceivingHelpDesk/ask/index.php" "SERVER_ADMIN" => "you@example.com" "CONTEXT_DOCUMENT_ROOT" => "/home/runcloud/webapps/ReceivingHelpDesk/" "CONTEXT_PREFIX" => "" "REQUEST_SCHEME" => "http" "DOCUMENT_ROOT" => "/home/runcloud/webapps/ReceivingHelpDesk/" "REMOTE_ADDR" => "172.69.7.208" "SERVER_PORT" => "80" "SERVER_ADDR" => "127.0.0.1" "SERVER_NAME" => "receivinghelpdesk.com" "SERVER_SOFTWARE" => "Apache/2.4.63 (Unix) OpenSSL/1.1.1f" "SERVER_SIGNATURE" => "" "LD_LIBRARY_PATH" => "/RunCloud/Packages/apache2-rc/lib" "PATH" => "/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin" "HTTP_COOKIE" => "_pk_id.64.7c30=9eb14b80eb440b99.1749422317.; _pk_ses.64.7c30=1; XSRF-TOKEN=eyJpdiI6Ikh1V2lQK2tVMXJGOTZUYVdNaGsvYlE9PSIsInZhbHVlIjoiektZNU1Fb2R1Y1RlN3QxUzNVbkxCbGdpSUFER3VyZk1OMzZVNTJOWFJBM0dnNkF3MEx5Q3BiWkw2Yk5WaytYbHU2eHF0WXBsTkg0a2phcGV6QWpVWFJVUCtxbzRyaitQQlk3M1dCRHRBMW11bURKQm04cTBrd0dudko2bTJWWFoiLCJtYWMiOiI5NzMzMzkwODc0OGYwN2YwNjdjOTliYTI0YTA2MGI2ZTNkNzc5YTQxMmU4MTM0ZjM5ZGRlNzhkNTdlNzdkZTE5IiwidGFnIjoiIn0%3D; askhelpdesk_session=eyJpdiI6IkpNVVV6YkRaZXlJQ2toQ21ESW9Uenc9PSIsInZhbHVlIjoiYVY2eEo0T2tuR1lMQzBleGdtbS9YMlJHakhuU2dqc01pQ3ZBNVVYdGJ6RS9JSDVlNy9wemNSWUt0aGw5K1BENko3N2ZLWGsyYlFoOGViaE1ZZmx1VEpwS3I2Ryt4OG54dzBXQWxCYjFpRE9wSlpKSDFXaVo0V3hyaVhJbytDcWciLCJtYWMiOiIyN2I2YWYxNzBiNTc3NGU5MWE1ZDVkNDM2ZmZjOTA4YjlmMWU4ODk0MThkMmU2YmNhYTIwZGMzZWQ1Yzc1YTMxIiwidGFnIjoiIn0%3D_pk_id.64.7c30=9eb14b80eb440b99.1749422317.; _pk_ses.64.7c30=1; XSRF-TOKEN=eyJpdiI6Ikh1V2lQK2tVMXJGOTZUYVdNaGsvYlE9PSIsInZhbHVlIjoiektZNU1Fb2R1Y1RlN3QxUzNVbkxCb" "HTTP_CF_IPCOUNTRY" => "US" "HTTP_CF_CONNECTING_IP" => "216.73.216.192" "HTTP_CDN_LOOP" => "cloudflare; loops=1" "HTTP_SEC_FETCH_MODE" => "navigate" "HTTP_SEC_FETCH_SITE" => "none" "HTTP_ACCEPT" => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" "HTTP_USER_AGENT" => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" "HTTP_UPGRADE_INSECURE_REQUESTS" => "1" "HTTP_SEC_CH_UA_PLATFORM" => ""Windows"" "HTTP_SEC_CH_UA_MOBILE" => "?0" "HTTP_SEC_CH_UA" => ""Chromium";v="130", "HeadlessChrome";v="130", "Not?A_Brand";v="99"" "HTTP_CACHE_CONTROL" => "no-cache" "HTTP_PRAGMA" => "no-cache" "HTTP_ACCEPT_ENCODING" => "gzip, br" "HTTP_CF_RAY" => "94cbe19f0badf92b-ORD" "HTTP_PRIORITY" => "u=0, i" "HTTP_SEC_FETCH_DEST" => "document" "HTTP_SEC_FETCH_USER" => "?1" "HTTP_CF_VISITOR" => "{"scheme":"https"}" "HTTP_CONNECTION" => "close" "HTTP_X_FORWARDED_PROTO" => "https" "HTTP_X_FORWARDED_FOR" => "216.73.216.192, 172.69.7.208" "HTTP_X_SERVER_ADDR" => "154.12.239.204" "HTTP_HOST" => "receivinghelpdesk.com" "HTTPS" => "on" "REDIRECT_STATUS" => "200" "REDIRECT_HTTPS" => "on" "FCGI_ROLE" => "RESPONDER" "PHP_SELF" => "/ask/index.php" "REQUEST_TIME_FLOAT" => 1749422325.6234 "REQUEST_TIME" => 1749422325 ]
        request_cookies
        0 of 0
        array:4 [ "_pk_id_64_7c30" => null "_pk_ses_64_7c30" => null "XSRF-TOKEN" => "YIZbHgxfoG3uECSjkQip0kCmTrYX0szySBH9DUEt" "askhelpdesk_session" => "hXBiCewnzqDRd3Zg82gU64UrYywfwB1wyyWYyiDp" ]
        response_headers
        0 of 0
        array:7 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "private, must-revalidate" ] "date" => array:1 [ 0 => "Sun, 08 Jun 2025 22:38:45 GMT" ] "pragma" => array:1 [ 0 => "no-cache" ] "expires" => array:1 [ 0 => -1 ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IkpKczR0Mmdnc3B3TVhSQVJwMlgrUEE9PSIsInZhbHVlIjoiTWJHZjZQNDY3cS9sUHNGUVd0RVhJczloNXFjZ3BwSUkycVYyS2VBVWNsUFZ3QWtrZTdNSFRuYTdyZmRCcDBSSVBHeElhc3dqUi9Ja2VwRWZmcEUvcTNHdjVEYXM1Z3BJYUM3OENtK05VNkRoc0hnQjZOK2pyVm13SXA0WHp4NDQiLCJtYWMiOiI0OWJiMGQ1OTYxZGQyNDcxZmM1NjFjYTBmZTlmMWE2YmE4YzAxZWI1ZjY5OWZjNWNmYjkxZWUyODEyZWE0MmY1IiwidGFnIjoiIn0%3D; expires=Mon, 09-Jun-2025 00:38:46 GMT; Max-Age=7200; path=/; samesite=laxXSRF-TOKEN=eyJpdiI6IkpKczR0Mmdnc3B3TVhSQVJwMlgrUEE9PSIsInZhbHVlIjoiTWJHZjZQNDY3cS9sUHNGUVd0RVhJczloNXFjZ3BwSUkycVYyS2VBVWNsUFZ3QWtrZTdNSFRuYTdyZmRCcDBSSVBHeElhc" 1 => "askhelpdesk_session=eyJpdiI6IjBzL1NNMFM5cU5ld0tGcGJjUzZJRUE9PSIsInZhbHVlIjoiclgxZXV6ZHBUaHFxVGpEbXJ5cG85SWp4cjk1N0tjbkVrcmNJTGdpc2kzbWRoZDJmVzh4ZFJBRGFnSHg1anFHVEYyenp0ZFQwVmx6ZDVxWTBLN0I4TFA1dUxHS1Z5aWJzRjJ3WXB3TTN1cS95K3JJcURjc2YzdVdXSzZnaS8vU0siLCJtYWMiOiIyOWExODFmYzY1ZGJhZjBlM2M4MjZjMWFhMDM0OWFlY2RjYmUyN2MyYjliZWVmZWQwMmU3NTE4NjAyZjEzY2YyIiwidGFnIjoiIn0%3D; expires=Mon, 09-Jun-2025 00:38:46 GMT; Max-Age=7200; path=/; httponly; samesite=laxaskhelpdesk_session=eyJpdiI6IjBzL1NNMFM5cU5ld0tGcGJjUzZJRUE9PSIsInZhbHVlIjoiclgxZXV6ZHBUaHFxVGpEbXJ5cG85SWp4cjk1N0tjbkVrcmNJTGdpc2kzbWRoZDJmVzh4ZFJBRGFnSHg1anFH" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IkpKczR0Mmdnc3B3TVhSQVJwMlgrUEE9PSIsInZhbHVlIjoiTWJHZjZQNDY3cS9sUHNGUVd0RVhJczloNXFjZ3BwSUkycVYyS2VBVWNsUFZ3QWtrZTdNSFRuYTdyZmRCcDBSSVBHeElhc3dqUi9Ja2VwRWZmcEUvcTNHdjVEYXM1Z3BJYUM3OENtK05VNkRoc0hnQjZOK2pyVm13SXA0WHp4NDQiLCJtYWMiOiI0OWJiMGQ1OTYxZGQyNDcxZmM1NjFjYTBmZTlmMWE2YmE4YzAxZWI1ZjY5OWZjNWNmYjkxZWUyODEyZWE0MmY1IiwidGFnIjoiIn0%3D; expires=Mon, 09-Jun-2025 00:38:46 GMT; path=/XSRF-TOKEN=eyJpdiI6IkpKczR0Mmdnc3B3TVhSQVJwMlgrUEE9PSIsInZhbHVlIjoiTWJHZjZQNDY3cS9sUHNGUVd0RVhJczloNXFjZ3BwSUkycVYyS2VBVWNsUFZ3QWtrZTdNSFRuYTdyZmRCcDBSSVBHeElhc" 1 => "askhelpdesk_session=eyJpdiI6IjBzL1NNMFM5cU5ld0tGcGJjUzZJRUE9PSIsInZhbHVlIjoiclgxZXV6ZHBUaHFxVGpEbXJ5cG85SWp4cjk1N0tjbkVrcmNJTGdpc2kzbWRoZDJmVzh4ZFJBRGFnSHg1anFHVEYyenp0ZFQwVmx6ZDVxWTBLN0I4TFA1dUxHS1Z5aWJzRjJ3WXB3TTN1cS95K3JJcURjc2YzdVdXSzZnaS8vU0siLCJtYWMiOiIyOWExODFmYzY1ZGJhZjBlM2M4MjZjMWFhMDM0OWFlY2RjYmUyN2MyYjliZWVmZWQwMmU3NTE4NjAyZjEzY2YyIiwidGFnIjoiIn0%3D; expires=Mon, 09-Jun-2025 00:38:46 GMT; path=/; httponlyaskhelpdesk_session=eyJpdiI6IjBzL1NNMFM5cU5ld0tGcGJjUzZJRUE9PSIsInZhbHVlIjoiclgxZXV6ZHBUaHFxVGpEbXJ5cG85SWp4cjk1N0tjbkVrcmNJTGdpc2kzbWRoZDJmVzh4ZFJBRGFnSHg1anFH" ] ]
        session_attributes
        0 of 0
        array:4 [ "_token" => "YIZbHgxfoG3uECSjkQip0kCmTrYX0szySBH9DUEt" "_previous" => array:1 [ "url" => "https://receivinghelpdesk.com/ask/4-by-4-matrix" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]