Here the answer is b) i.e. A2 = I
let's see each options,
A) A is a zero matrix - FALSE
Zero matrices are those matrices which when multiplied to any other matrix generate a zero matrix.
ref :-https://youtu.be/LOf8bfjiLow
B) A(square) = Identity matix - TRUE
simply multiply the given matrix to itself you will get identity matrix.
C) A(inverse) does not exist - FALSE
Generally A(inverse) = (Adjoint of A) / determinant(A)
here the determinant is 1 which is not equal to zero
hence A(inverse) is exist.
D) A= (-1) I, where I is a Unit matrix - FAlSE
Even though it is looking like the identity matrix which is written after applying some row interchange operations. But it is not the Identity matix because identity matrix has the property that when ever we multiply any matrix let's say X then the result will be matrix X itself (as it is).
But if we multiply it with any other matrix then the resultant matrix is not as it is which it was initially, there are some rows which interchange itself in the resultant matrix.
Hence B) is correct answer.