If A is an n squared matrix then show that $AA'$ and $A'A$ are Symmetric

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Vedantu Content Team · Accepted Answer

Hint-Use matrix propertiesAny matrix is said to be symmetric if and only if:$ 	o $The matrix is a square matrix and$ 	o $The transpose of the matrix must be equal to itself.Then here we know that the given Matrix A is a square matrix then the transpose of is A i.e. $A'$is also a square matrix. Here we know that $A'$ is our transpose matrix.Proof:$(AA')' = (A')'(A)$ [By using reversible law]$(AA'$$)'$$ = AA'$      $[\because (A')' = A]$$(AA'$$)'$$ = AA'$By using matrix properties we can say that $ = AA'$ is symmetricSimilarly if  $AA'$ is symmetric then $A'A$is also symmetricHence we proved that for any n squared matrix  $A'A$  and  $AA'$  are symmetricNOTE: This problem can also be solved directly by stating the matrix properties as they have already proved, for which statement is enough.

If A is an n squared matrix then show that $AA'$ and $A'A$ are Symmetric

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If A is an n squared matrix then show that $AA'$ and $A'A$ are Symmetric

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