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Let A and B are two non-singular square matrices, $A ^T$ and $B^T$ are the transpose matrices of A and B respectively, then which of the following is correct
Option: 1
$B^T$ AB is symmetric matrix if and only if A is symmetric

Option: 2
$B^T$ AB is symmetric matrix if and only if B is symmetric

Option: 3
   $B^T$ AB is skew symmetric matrix for every matrix A



Option: 4
    $B^T$ AB is skew symmetric matrix if B is skew symmetric

Answers (1)

best_answer

Property of Transpose -

$\left ( \alpha A \right ){}'= \alpha A{}'$

$(AB)^{'} =B^{'}A^{'}$

- wherein

$\alpha$ being scalar ; $A{}'$ is transpose of A

Symmetric matrix -

If $A=\left [ a_{ij} \right ]$ and $a_{ij}=a_{ji}$ for all $i$ and $j$

- wherein

Skew symmetric matrix -

If $A=\left [ a_{ij} \right ]$ and $a_{ij}=-a_{ji}$ for all $i$ and $j$

- wherein

$(B^T AB ) ^ T = B^T A^T ( B^T )^T = B^T A^T B \\\\ = B^T AB$

if A is symmetric

$\therefore$ $B^T AB$ is symmetric if A is symmetric

Also $(B^T AB ) ^ T = B^T A^T B = ( - B) A ^ T B$

$\therefore$ $B^T AB$ is not skew symmetric if B is skew

Posted by

shivangi.bhatnagar

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