Question

if a and b are square matrices of the same order such that ab = ba ,then show that (a+b)³ = a³+3a²b+3ab²+b³.​

Answer 1

Answer:

Given A and B are square matrices

A

2

=A,B

2

=B,AB=BA=O

(A+B)

2

=(A+B)(A+B)

(A+B)

2

=A

2

+AB+BA+B

2

⇒(A+B)

2

=A+B (using given properties)

(AB)

2

=(AB)(AB)=O

(A−B)

2

=(A−B)(A−B)

(A−B)

2

=A

2

+B

2

−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