(a) Explain why for the product A^2 to make sense, that it is necessary that A be a square matrix.
(b) Begin with (A - I)(A + I) = 0. Using the properties of multiplication and addition, show that any square matrix A that satisfies the equation (A - I)(A + I) = 0 also satisfies the condition A^2 = I. Justify each algebraic simplification with the appropriate property.
(c) Find a 2 x 2 matrix A such that A2 = I but A ≠ ±I.
Again, in the real number system if ab = 0 then either a = 0, b = 0 or both. This is
another property that does not hold in matrix multiplication.
The squaring of a matrix is investigated.