How can we prove that
where A and B are two matrices of same order and T represents transpose of matrix.
Two matrices are equal if they are of same order and their corresponding elements are equal. In the same way, if we can prove that both sides of the equation
have same order and their corresponding elements are equal then it means that the given equation is true.
Lets prove first that both the sides of equation have same order:
Suppose order of both A and B is m x n.
Order of A+ B would also be m x n
Order of
would be n x m {According to definition of transpose of Matrix}
Now, check order of right hand side of equation
Order of both
and
is n x m.
Order of
would also be n x m.
Therefore order of both the sides of equation is same.
Now, we need to prove equal corresponding elements.
element of
is equal to
element of
=
element of A +
element of B
=
element of
+
element of
=
element of
It means that corresponding elements of
and
are also equal and we have proved that their order is also same. Therefore,
. Similarly, following the same method, we can prove property