Question

A: $AB=A$ and $BA=B\Rightarrow {{A}^{n}}+{{B}^{n}}=A+B$.
                    R: $AB=A$ and $BA=B\Rightarrow A$ and $B$ are idempotent.
A.Both A and R are true and R is the correct explanation to A.
B.Both A and R are true but R is not the correct explanation for A.
C.A is true R is false.
D.A is false R is true

Answer 1

Hint: Here, to prove A we have to apply the principle of mathematical induction. That is first prove that the statement is true for n = 1 and then assume that the statement is true for n = k and then prove that it is true for n = k+1. From the induction part itself you can prove that R is true.

Complete step by step answer:
Here, we are given that $AB=A$ and $BA=B$.
Now, we have to prove that ${{A}^{n}}+{{B}^{n}}=A+B$.
Here, we can apply the principle of mathematical induction.
We know that for the principle of mathematical induction first we have to prove that the statement is true for n =1, then we have to assume that it is also true for n = k, k is any integer and prove that it is true for n = k+1.
For n = 1 we have $A+B=A+B$
Therefore, we can say that $BA=B\Rightarrow {{A}^{n}}+{{B}^{n}}=A+B$ is true for n=1.
Now consider n = 2, i.e. ${{A}^{2}}+{{B}^{2}}$.
We have, $AB=A$ and $BA=B$. Hence, we can write:
$\begin{align}
  & A(BA)=A \\
 & ABA=A \\
 & (AB)A=A \\
\end{align}$
Now, by substituting $AB=A$ we get:
$AA=A$
${{A}^{2}}=A$ ….. (1)
Similarly we can write:
$\begin{align}
  & BA=B \\
 & B(AB)=B \\
 & BAB=B \\
 & (BA)B=B \\
\end{align}$
Now, by substituting $BA=B$, we will get:
$BB=B$
${{B}^{2}}=B$ ….. (2)
Now, from equation (1) and equation (2), we can write:
${{A}^{2}}+{{B}^{2}}=A+B$
Hence, we can say that $BA=B\Rightarrow {{A}^{n}}+{{B}^{n}}=A+B$ is true for n = 2.
Now, we can assume that it is true for n = k. Next, we have to prove that it is true for k+1.
For n = k, we have:
${{A}^{k}}+{{B}^{k}}=A+B$
Now, we can write:
${{A}^{k+1}}+{{B}^{k+1}}={{A}^{k}}A+{{B}^{k}}B$ ….. (3)
Here, since ${{A}^{2}}=A$ and ${{B}^{2}}=B$, we can write:
${{A}^{k}}=A$ and ${{B}^{k}}=B$
Hence, our equation (3) becomes:
$\begin{align}
  & {{A}^{k+1}}+{{B}^{k+1}}=AA+BB \\
 & {{A}^{k+1}}+{{B}^{k+1}}={{A}^{2}}+{{B}^{2}} \\
\end{align}$
We already proved that ${{A}^{2}}+{{B}^{2}}=A+B$. Therefore, we can say that:
${{A}^{k+1}}+{{B}^{k+1}}=A+B$
Hence by mathematical induction we can write:
${{A}^{n}}+{{B}^{n}}=A+B$
Therefore, we can say that $A:AB=A$ and $BA=B\Rightarrow {{A}^{n}}+{{B}^{n}}=A+B$.
Next, we can consider $R:AB=A$ and $BA=B$.
Now, by equation (1) and equation (2) we have:
${{A}^{2}}=A$ and ${{B}^{2}}=B$
We know that an idempotent matrix is a square matrix which when multiplied by itself yields itself. That is, if A is a matrix then ${{A}^{2}}=A$
Hence, by the definition we can say that A and B are idempotent.
Thus we can write, R: $AB=A$ and $BA=B\Rightarrow A$ and $B$ are idempotent.
Therefore, we can say that both A and R are true and R is the correct explanation to A.
Hence, the correct answer for this question is option (a).

Note: Here, you have to use the principle of mathematical induction. Mathematical induction is a technique of proving a statement, theorem or formula which is thought to be true; for each and every natural number n. By generalising this in the form of principle which we would use to prove any mathematical statement is the principle of mathematical induction.