Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0000 361232 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0493 1211720 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0493 1211720 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1188 4132528 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1201 4160512 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1207 4178224 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1209 4208744 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1220 4231464 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1220 4231464 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1746 4903128 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1781 5095280 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1786 5116208 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.1902 5327376 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.1902 5327632 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182 Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; JCommentsACL has a deprecated constructor in /var/www/astarmathsandphysics/components/com_jcomments/classes/acl.php on line 17 Call Stack: 0.0000 361232 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0493 1211720 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0493 1211720 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1188 4132528 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1201 4160512 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1207 4178224 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1209 4208744 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1220 4231464 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1220 4231464 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1746 4903128 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1781 5095280 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1786 5116208 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3261 13753640 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.3263 13754040 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.3263 13754040 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.3265 13762288 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.3267 13764024 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.3274 13797072 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.3274 13797472 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.3275 13797552 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

Proof That Multiplying a Row or Column of a Matrix By a Constant Multiplies the Determinant by that Constant

Theorem
Multiplying a row or column of a square matrix by a constant multiplies the determinant of the matrix by that constant.
Proof
For a matrix  
\[A\]
  with elements  
\[(a_{ij})\]
  the determinant is
\[\left| A \right| = \sum_{\sigma} sgn ( \sigma ) a_{1k_1} a_{2 k_2} ... a_{nk_n}\]
.
where  
\[sgn (\sigma )\]
  is the parity of the  
\[k_1 , \: k_2 ,..., \: k_n \]
ie the number of transitions required to bring  
\[k_1 , \: k_2 ,..., \: k_n \]
  to the order 
\[1, \: 2, \: ..., \: n \]
.
If an even number of transitions is required then  
\[sgn( \sigma )=1 \]
  and if an odd number of transitions is required then  
\[sgn( \sigma )=-1 \]
.
Suppose a row of  
\[A\]
  is multiplied by  
\[\alpha\]
  to give a matrix  
\[A.\]
.
Every term in the determinant now includes a factor  
\[c\]
  and looks like  
\[sgn ( \sigma ) a_{1k_1} a_{2 k_2} ... ca_{jk_j} ... a_{nk_n}= c sgn ( \sigma ) a_{1k_1} a_{2 k_2} ... a_{nk_n}\]
.\[\left| A' \right| =c \sum_{\sigma} sgn ( \sigma ) a_{1k_1} a_{2 k_2} ... a_{nk_n}= c \left| A \right|\]
.
Since  
\[\left| A \right| = \left| A^T \right| \]
  from this result, multiplying a column by a constant also multiplies a determinant by the same constant.

Add comment

Security code
Refresh