Question

Let $A,B$ be two matrices of same order such that $AB=A$ and $BA=B.$ Then $(A+B{)}^{2020}$ equals
(Here, $I$ is the identity matrix of same order.)

• A. $2^{2019}(A+B)$
• B. $2020I$
• C. $2^{2020}I$
• D. $2^{2020}(A+B)$

Answer 1

The correct option is A $2^{2019}(A+B)$

Since $AB=A$ and $BA=B$
$\therefore ABA=AA\Rightarrow AB=A^2\Rightarrow A=A^2$
Similarly, $B=B^2$
Now,
$(A+B{)}^2=(A+B\left)\right(A+B)$
$\Rightarrow (A+B{)}^2=A^2+AB+BA+B^2$
$\Rightarrow (A+B{)}^2=2(A+B)$
Similarly, $(A+B{)}^3=2^2(A+B)$
$\therefore (A+B{)}^n=2^{n-1}(A+B)$
Hence, $(A+B{)}^{2020}=2^{2019}(A+B)$