JEE Main · 2006 · 30 Apr 2006 · Offline · Matrices

A

and

B

are square matrices of size

n \times n

such that

A^{2} - B^{2} = (A - B) (A + B)

, then which of the following will be always true?

Answer: (B)

Solution

Multiply out the right side keeping the order of the matrices:

(A - B) (A + B) = A^{2} + A B - B A - B^{2}

. The problem states this equals

A^{2} - B^{2}

, so

A^{2} - B^{2} = A^{2} + A B - B A - B^{2}

, which collapses to

A B - B A = 0

. The only condition guaranteed in every case is therefore that

A

and

B

commute,

A B = B A

JEE Main · 2006 · 30 Apr 2006 · Offline · Single correct option