Chapter 4.4, Problem 1ED

In addition to the commutative and zero properties, there are other significant differences between real number multiplication and matrix multiplication.

(A) In real number multiplication, the only real number whose square is $0$ is the real number $0\left(0^2=0\right)$. Find at least one $2\times 2$ matrix A with all elements nonzero such that $A^2=0$, where 0 is the $2\times 2$ zero matrix.

(B) In real number multiplication, the only nonzero real number that is equal to its square is the real number $1\left(1^2=1\right)$. Find at least one $2\times 2$ matrix B with all elements nonzero such that $B^2=B$.