We are given a matrix A and scalar k, we want to 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, if we can prove that both the sides of equation
have same order and their corresponding elements are equal then it means equation
is true.
Lets first prove that both the sides of equation
have same order.
Let order of A be m x n
Then, Order of (kA) would also be m x n
Order of
would be n x m {According to definition of Transpose of Matrix}
Now, find out order of right hand side of equation
We have supposed that order of A is m x n.
Then, Order of
is equal to n x m
Order of
would also be equal to n x m
Therefore,
and
have same order equal to n x m. Now, lets prove that their corresponding elements are equal.
element of
=
element of
= k .
element of
=k .
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, equation
is true.
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