Question

15. If for two square matrices A and B of same order,
A2 = A, B2 = Band (A + B)2 = A + B, then prove
that AB = BA = O.​

Answer 1

Answer:

(A+B)2 =(A−B)2

Step-by-step explanation:

Given A and B are square matrices

A2 =A, B2 =B, AB =BA =O

(A+B)2 =(A+B)(A+B)

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

⇒(A+B)2 =A + B (using given properties)

(AB)2 =(AB)(AB) =O

(A−B)2 =(A−B)(A−B)

(A−B)2 =A2 + B2 − AB − BA

⇒(A−B)2 =A + B (using given properties)

∴(A+B)2 =(A−B)2

,

(AB)2=0, and (A−B)2=A+B

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