Question

If matrix B^2=B^3, determine all the values of B^2.

I know it’s a very general question, but can anyone at least give me some idea as to how to approach this question?

Thank you!

Answer 1

We know that:

A nilpotent matrix is a square matrix A such that{\displaystyle N^{k}=0\,} Aⁿ = 0 for some positive integer n.

An idempotent matrix is a matrix which, when multiplied by itself, yields itself.

If matrix B² =B³, the B has to be square matrix. Then, there are following possibilities:

1. B is singular. If so, B is either a zero matrix or B is a nilpotent matrix with index 2 so that B² = 0. In such a case, B³ = B².B = 0.B = 0 = B². Alternatively, B can be an idempotent matrix so that B² = B. Then B³ = B² .B = B.B = B².

2. B is non-singular. Since B³ =B², on multiplying to the right by B^-1 , we get B² = B. Thus, B is an idempotent matrix.