Final answer:
To determine if a matrix is nilpotent and find the matrix exponential
Step-by-step explanation:
To show that a matrix A is nilpotent, we need to find a positive integer k such that A^k = 0, where 0 is the zero matrix.
To find the matrix exponential, we can use the formula e^A = I + A + (A^2)/2! + (A^3)/3! + ... + (A^n)/n!, where I is the identity matrix.
For the given problems, you need to follow the steps to prove that A is nilpotent and then use the formula to find the matrix exponential.