Show that A A and A A are both symmetric matrices for any matrix A.

Let P = A A = ( A A ) = A ( A ) = A A = P Therefore, A A is symmetric matrix for any matrix A. Similarly, let Q = A A = ( A A ) = ( A ) (A) = A ( A ) = Q Therefore, A A is symmetric matrix for any matrix A.

Matrices — Class 12 Maths Solution

exemplar sa SA NCERT,Exemp,Q.no.29,Page 56

Question

Show that ${A^\prime }A$ and $A{A^\prime }$ are both symmetric matrices for any matrix A.

Step-by-step Solution

Let $P = {A^\prime }A$
$\therefore$ ${P^\prime } = {\left( {A{A^\prime }} \right)^\prime }$

$= {A^\prime }{\left( {{A^\prime }} \right)^\prime }$

$= {A^\prime }A = P$

Therefore, ${A^\prime }A$ is symmetric matrix for any matrix A.

Similarly, let $Q = A{A^\prime }$

$\therefore$ ${Q^\prime } = {\left( {A{A^\prime }} \right)^\prime } = {\left( {{A^\prime }} \right)^\prime }{(A)^\prime }$

$= A{\left( {{A^\prime }} \right)^\prime } = Q$

Therefore, $A{A^\prime }$ is symmetric matrix for any matrix A.

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