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