We are given matrix A then how can we prove that
. where T represents transpose of Matrix.
Two matrices are said to be equal if they have same order and their corresponding elements are equal. Similarly, to prove
. we will have to prove that both the sides have same order and both the sides have equal corresponding elements.
Lets first prove that
and
have same order.
Let order of Matrix A be m x n
Then, order of
would be n x m {According to definition of transpose of matrix}
Then, order of
would be again m x n {According to definition of transpose of matrix}
Therefore, order of both
and
is same.
Lets prove now that both the sides of equation
have equal corresponding elements.
element of
=
element of
{According to definition of transpose of matrix}
=
element of
{According to definition of transpose of matrix}
Therefore,
element of
=
element of
Therefore, we can say that corresponding elements of
and
are equal. We have already proved that their order is same.
Therefore, we can say that equation
is true.
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