Question

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

Answer 1

Hint-Use matrix properties
Any matrix is said to be symmetric if and only if:
$ \to $The matrix is a square matrix and
$ \to $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 symmetric
Similarly if $AA'$ is symmetric then $A'A$is also symmetric
Hence we proved that for any n squared matrix $A'A$ and $AA'$ are symmetric

NOTE: This problem can also be solved directly by stating the matrix properties as they have already proved, for which statement is enough.