Please solve complete problem. I need complete step by step solution. Book: Fundamental of differential equation

2. Let cx denote the column of a matrix . Write down a matrix A withe; =[0 0 =2]T ,ca=[101]",c5=[012]". Recall that T denotes the transpose. {a) Without computing the characteristic polynomial of A, determine if the following vectors are eigenvectors of A. If ves, determine the correspond- ing eigenvalue. The vectors are [I =1 1]7, [1 1 1)" and [1 2 4]". (b) Use (a} or some other method to write down a fundamental matrix for the differential equation x'(#) = Ax(t). () Solve the initial value problem x'(t) = Ax(t), x(0) = [1 0 1]". (d) Give an example of a 3 x 1 non-constant vector that is not a solution of the initial value problem in (c). Prove that it is not a solution