#idempotent #matrix

Question

If Matrix A and B are of same size and AB=B and BA=A. Then the value of $(A+B)^{5}$ is _____________?

Comments

a,b denotes what?

Answer 1

Given: AB=B and BA=A

lets break the question as :

(A+B)⁵ = (A+B)² * (A+B)² * (A+B)

            =(A²+B²+AB+BA)(A²+B²+AB+BA)(A+B)...........@1

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(Since (A+B)²= (A²+AB+BA+B²) = (A*A + AB+BA+B*A)

                      =(A*BA+AB+BA+B*AB) = ((A*B)A+AB+BA+(B*A)B)

                      =(BA+AB+BA+AB) = 2(A+B)...............@2

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Puting value of equation @2 in @1 we get :->

                      =2(A+B)*2(A+B)*(A+B) = 4(A+B)²*(A+B) = 8(A+B)(A+B)

                      =16(A+B) _answer