1) is there a math object or number (any type; quaternion or its further generalization, for example) that becomes zero when some operation is applied to the object using itself? The object must not be zero. also, when does such object not become zero when operated on it by another object?
2) is there an object that becomes zero when operated(multiplied) by another object?
3) is there any math object or matrix that satisfies the following property (I ask for the object that satisfies both constraints):
a)$ab+ba=0$ where a and b are math objects or matrices
b) $A^2=B^2$
for all objects I assume nonzero.