Final answer:
The given matrix A = [1 -1; 1 -1] is nilpotent since its square results in the zero matrix. Consequently, the matrix exponential e^At simplifies to I+At for nilpotent matrices, in this case resulting in [1+t -t; t 1-t].
Step-by-step explanation:
A matrix is said to be nilpotent if it raises to some power resulting in the zero matrix. To show that a given matrix A is nilpotent, we must find such a power where An equals the zero matrix. For the matrix A = [1 -1; 1 -1], let's calculate A2 to see if it becomes the zero matrix.
Multiplying the matrix by itself, we get A2 = [1 -1; 1 -1] [1 -1; 1 -1] which results in [0 0; 0 0]. Thus, the matrix A is nilpotent because A2 = 0.
With this information, we can find the matrix exponential eAt which is defined as the infinite series I + At + A2t2/2! + .... However, since A2 = 0, all higher powers of A will also be zero. This simplifies the matrix exponential to eAt = I + At, where I is the identity matrix.
Thus, eAt = [1 0; 0 1] + t [1 -1; 1 -1] = [1+t -t; t 1-t]. This exemplifies the process of finding the matrix exponential for a nilpotent matrix, which has significant applications in differential equations and linear systems theory.