- #1

vcsharp2003

- 897

- 177

- Homework Statement
- 4. If ##A## and ##B## are two matrices such that ##AB = B## and ## BA = A##, then ##A^2

+ B^2## equals:

(a) ##A + B##

(b) ##2BA##

(c) ##2AB##

(d)##BA##

- Relevant Equations
- ##AI = IA = A## where ##I## is identity matrix and ##A## is any square matrix whose product with identity matrix is defined

Since ##AB = B##, so matrix ##A## is an identity matrix.

Similarly, since ##BA = A## so matrix ##B## is an identity matrix.

Also, we can say that ##A^2 = AA=IA= A## and ##B^2 = BB=IB= B##.

Therefore, ##A^2 + B^2 = A + B## which means (a) is a correct answer.

Also we can say that ##A^2 + B^2 = I^2 + I^2 = II + II = AB + AB = 2AB##,

and that ##A^2 + B^2 = I^2 + I^2 = II + II = BA + BA = 2BA##. From these conclusions, it also follows that (b) and (c) are correct answers.

Similarly, since ##BA = A## so matrix ##B## is an identity matrix.

Also, we can say that ##A^2 = AA=IA= A## and ##B^2 = BB=IB= B##.

Therefore, ##A^2 + B^2 = A + B## which means (a) is a correct answer.

Also we can say that ##A^2 + B^2 = I^2 + I^2 = II + II = AB + AB = 2AB##,

and that ##A^2 + B^2 = I^2 + I^2 = II + II = BA + BA = 2BA##. From these conclusions, it also follows that (b) and (c) are correct answers.

*Thus, according to me (a),(b) and (c) are correct answers. But the correct answer is given as (a) only.*