if A, B are two idempotent matrices and AB=BA=0 then (A+B) is

(a) Scalar matrix
(b) diagonal matrix
(c) Nilpotent matrix
(d) idempotent matrix

Question

if A, B are two idempotent matrices an​d AB=BA=0 then
(A+B) is
(a) Scalar matrix
(b) diagonal matrix
(c) Nilpotent matrix
(d) idempotent matrix

Lovina Kansal · Accepted Answer

Dear student
Since A and B are idempotent, we haveA2=A and B2=BNow (A+B)2=A+BA+B=A+BA+A+BB    Distributive law=A2+BA+AB+B2   Distributive law=A+BA+AB+B   ∵A and B are idempotent=A+B+AB+BA    
(1)Sine AB=BA=0so, eq(1) becomesA+B2=A+BThis implies A+B is idempotent
Regards

if A, B are two idempotent matrices an​d AB=BA=0 then (A+B) is (a) Scalar matrix (b) diagonal matrix (c) Nilpotent matrix (d) idempotent matrix

if A, B are two idempotent matrices and AB=BA=0 then (A+B) is

(a) Scalar matrix
(b) diagonal matrix
(c) Nilpotent matrix
(d) idempotent matrix