# Prove the following A(rB)= r(AB)= (rA)B where r and s are real numbers and A and B are matrices

Question
Matrices
Prove the following
A(rB)= r(AB)= (rA)B
where r and s are real numbers and A and B are matrices

2021-02-15
Given that A and B are matrices and r is a real number.
Let us consider the matrix A is of order
$$\displaystyle{m}\times{n}$$
and let us consider the matrix B is of order
$$\displaystyle{n}\times{m}$$
$$\displaystyle{A}{\left({r}{B}\right)}={A}_{{{m}\times{n}}}{\left({r}{B}\right)}_{{{n}\times{m}}}$$
Hence the result for A(rB) will be of order
$$\displaystyle{m}\times{m}$$
$$\displaystyle{\left({r}{A}\right)}{B}={\left({r}{A}\right)}_{{{m}\times{n}}}{B}_{{{n}\times{m}}}$$
Hence the result for (rA)B will be of order
$$\displaystyle{m}\times{m}$$
Since the results for the above three is of order
$$\displaystyle{m}\times{m}$$
Therefore A(rB)= r(AB)= (rA)B

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