We are given with a matrix A and two scalars k1 and k2, how can we prove that
?
Two matrices are equal if they have same order and their corresponding elements are equal.
Similarly, if we can prove that
and
have same order and their corresponding elements are equal then it means equation
is true.
Lets first prove that
and
have same order.
Let order of matrix A be m x n
Order of
would also be m x n because
is just a scalar.
Now lets calculate order of right hand side of equation
.
We have supposed that order of A is m x n
Order of
would be m x n and order of
would also be equal to m x n.
Order of
would be m x n
Therefore,
and
have same order.
Now, lets prove that corresponding elements of
and
are equal.
element of
=
.
element of matrix A
=
.
element of matrix A +
.
element of matrix A
=
element of matrix
+
element of matrix
=
element of matrix (
Therefore, we can say that corresponding elements of
and
are equal. We have already proved that they have same order. Hence, equation
is true.
[…] 2. ({k_1} + {k_2})A = {k_1}A + {k_2}A […]