Question

If $A$ and $B$ be 2 square matrices of the third order such that $AB=A,BA=B$, then $(A+B{)}^7$ is:

• A. $7(A+B)$
• B. $35(A+B)$
• C. $64(A+B)$
• D. None of these

Answer 1

Answer: (C)

$64(A+B)$

$AB=A$

$\to ABA=A^2$

$\to AB=A^2$

$\to A=A^2$

also we can see

$B^2=$ B
therefore,
$(A+B{)}^2=A^2+B^2+AB+BA=2(A+B)$
$(A+B{)}^6=2^3(A+B{)}^3$
$(A+B{)}^7=2^3(A+B{)}^4$
$=2^3\left(\right(A+B{)}^2{)}^2$
$=2^5(A+B{)}^2$
$=2^6(A+B)$
$(A+B{)}^7=64(A+B)$