Higher Mathematics for Science, Technology and Engineering

Theory of Determinants and Matrices and their Applications in Linear Equation Theory

Author(s): S. G. Ahmed

Pp: 180-225 (46)

DOI: 10.2174/9781681082813116010006

* (Excluding Mailing and Handling)

Abstract

Determinants and matrices are very important subjects that are widely found in different branches and topics in mathematics. The present chapter deals with these two topics in some brief details starting the preliminary definitions of determinants and matrices, their properties. The linear system of equations from its mathematical constitution till different methods of solutions is also discussed in brief details.


Keywords: Cofactors, determinants, elementary matrices, eigenvectors, Laplace expansion, minors, matrix operations.

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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
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      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 2002 and `b`.`banner_from_date` <= 1740525702 and `b`.`banner_to_date` >= 1740525702 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.43msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 2002
      • 3: 1740525702
      • 4: 1740525702
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='e92c8ff52890132c44598f5d361fe144'
      250μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Keywords
    7Keywords.php#?
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        Crvif680I243hUfSvBtfJY2XCJ91GyPWIretNiI4
        uc_cart_id
        e92c8ff52890132c44598f5d361fe144
        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/9329" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /chapter/9329
        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:17 [ "x-https" => array:1 [ 0 => "1" ] "host" => array:1 [ 0 => "alpha.eurekaselect.com" ] "priority" => array:1 [ 0 => "u=0, i" ] "cookie" => array:1 [ 0 => "wayfindertarget=eyJpdiI6InlxT0xEWmtRdEF5eWtMSFE2bm41OUE9PSIsInZhbHVlIjoianNLQi9uQnUvcHhXVWg4cm8ybEpCcVhKMUgrWkNHVU9GajJNRW9MNFM1bz0iLCJtYWMiOiIzZGI2YjkxZmMwNzI1MTIzN2Q5ZjQ5ZTZlMzgwNWZlYzdmZWQ0MDU4NDMxMDZiMTMxY2EzYjRlZGM2NjM1M2EyIiwidGFnIjoiIn0%3Dwayfindertarget=eyJpdiI6InlxT0xEWmtRdEF5eWtMSFE2bm41OUE9PSIsInZhbHVlIjoianNLQi9uQnUvcHhXVWg4cm8ybEpCcVhKMUgrWkNHVU9GajJNRW9MNFM1bz0iLCJtYWMiOiIzZGI2YjkxZmMwNzI1" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?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 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] ]
        request_cookies
        0 of 0
        array:1 [ "wayfindertarget" => null ]
        response_headers
        0 of 0
        array:5 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "no-cache, private" ] "date" => array:1 [ 0 => "Tue, 25 Feb 2025 23:21:42 GMT" ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6InpteXlzaWhZT0swYVE1OVllYVNSQ3c9PSIsInZhbHVlIjoibkg1ckM2eUJTdEJXOTRLdERsMWlDVHR6YTJZRVdHWk9TMEUydlRIRFdHa3NYV0pXYUc4aFVVRUlxVGFURXRuNGl1Q0I0VUFPTEttS25BRTl0UUF1bTJlWHUvbXRMTktDRUlMYVN5QXkwYTlyTmV1aGlUUUVPY004Q1pMeFRHbnMiLCJtYWMiOiI3YmE2MTAwZmU3Y2IwZGM3YTM3MmYxMDk0MWFhMzU1YTAzOTJiODUyY2FlNGU0ODViNzIxZjk3ZmVlMGIzMjdhIiwidGFnIjoiIn0%3D; expires=Fri, 28 Feb 2025 11:21:42 GMT; Max-Age=216000; path=/; secure; samesite=laxXSRF-TOKEN=eyJpdiI6InpteXlzaWhZT0swYVE1OVllYVNSQ3c9PSIsInZhbHVlIjoibkg1ckM2eUJTdEJXOTRLdERsMWlDVHR6YTJZRVdHWk9TMEUydlRIRFdHa3NYV0pXYUc4aFVVRUlxVGFURXRuNGl1Q0I0V" 1 => "alphaeurekaselectnet_session=eyJpdiI6InNEakh6Q3cwcnU5TnZqVS9CU3BqWWc9PSIsInZhbHVlIjoienJCZjR3U1F6blN3cDZkK2FjQ0FXczR3TEpaMHJhS3JLczdnMFd0c09seVlTanh5Y1BZN2ZFMFR0aFI5M25jelN1KzA5VUJtclRrNGd6NE1JeW5TM1c5L1lnVVVEQkdjQUtLZ212R3hQYlI3MS8vdnhxYTFyRk1TdlFma2w1a3QiLCJtYWMiOiI3YjZhYmE3ZTdhODg4ODQwYjZjYTA4NjA4ZTI1ZmE4ZjBiNTE0MzRkODY0ZWE1ZGY3NWJkMjRlMDQ4MWU1YjFjIiwidGFnIjoiIn0%3D; path=/; secure; httponly; samesite=laxalphaeurekaselectnet_session=eyJpdiI6InNEakh6Q3cwcnU5TnZqVS9CU3BqWWc9PSIsInZhbHVlIjoienJCZjR3U1F6blN3cDZkK2FjQ0FXczR3TEpaMHJhS3JLczdnMFd0c09seVlTanh5Y1BZN2ZFMFR" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6InpteXlzaWhZT0swYVE1OVllYVNSQ3c9PSIsInZhbHVlIjoibkg1ckM2eUJTdEJXOTRLdERsMWlDVHR6YTJZRVdHWk9TMEUydlRIRFdHa3NYV0pXYUc4aFVVRUlxVGFURXRuNGl1Q0I0VUFPTEttS25BRTl0UUF1bTJlWHUvbXRMTktDRUlMYVN5QXkwYTlyTmV1aGlUUUVPY004Q1pMeFRHbnMiLCJtYWMiOiI3YmE2MTAwZmU3Y2IwZGM3YTM3MmYxMDk0MWFhMzU1YTAzOTJiODUyY2FlNGU0ODViNzIxZjk3ZmVlMGIzMjdhIiwidGFnIjoiIn0%3D; expires=Fri, 28-Feb-2025 11:21:42 GMT; path=/; secureXSRF-TOKEN=eyJpdiI6InpteXlzaWhZT0swYVE1OVllYVNSQ3c9PSIsInZhbHVlIjoibkg1ckM2eUJTdEJXOTRLdERsMWlDVHR6YTJZRVdHWk9TMEUydlRIRFdHa3NYV0pXYUc4aFVVRUlxVGFURXRuNGl1Q0I0V" 1 => "alphaeurekaselectnet_session=eyJpdiI6InNEakh6Q3cwcnU5TnZqVS9CU3BqWWc9PSIsInZhbHVlIjoienJCZjR3U1F6blN3cDZkK2FjQ0FXczR3TEpaMHJhS3JLczdnMFd0c09seVlTanh5Y1BZN2ZFMFR0aFI5M25jelN1KzA5VUJtclRrNGd6NE1JeW5TM1c5L1lnVVVEQkdjQUtLZ212R3hQYlI3MS8vdnhxYTFyRk1TdlFma2w1a3QiLCJtYWMiOiI3YjZhYmE3ZTdhODg4ODQwYjZjYTA4NjA4ZTI1ZmE4ZjBiNTE0MzRkODY0ZWE1ZGY3NWJkMjRlMDQ4MWU1YjFjIiwidGFnIjoiIn0%3D; path=/; secure; httponlyalphaeurekaselectnet_session=eyJpdiI6InNEakh6Q3cwcnU5TnZqVS9CU3BqWWc9PSIsInZhbHVlIjoienJCZjR3U1F6blN3cDZkK2FjQ0FXczR3TEpaMHJhS3JLczdnMFd0c09seVlTanh5Y1BZN2ZFMFR" ] ]
        session_attributes
        0 of 0
        array:5 [ "_token" => "Crvif680I243hUfSvBtfJY2XCJ91GyPWIretNiI4" "uc_cart_id" => "e92c8ff52890132c44598f5d361fe144" "_previous" => array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/9329" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]
        ClearShow all
        Date ↕MethodURLData
        #12025-02-25 23:21:42GET/chapter/9329112189