We recall the definition of eigenvalues and eigenvectors. For a square matrix A, if I is a real number and ver” is a non-zero vector such that the equation Αν = λν holds, then 1 is an eigenvalue and v is an eigenvector associated to 1. Note that the condition v 70 is important. In this question, we ask you to prove several properties of eigenvalues directly from this definition. In particular, you should NOT use and in fact are NOT allowed to determinants or characteristic polynomials at all in your working. The computer algorithms that compute eigenvalues look very different from the tech- niques presented in a first-year undergraduate class. Many of the algorithms rely on the following properties of eigenvalues: (a) If A is a square matrix and t is a real number, then 1 is an eigenvalue of A if and only if 1 +t is an eigenvalue of A+ tl. (Here I is the identity matrix.) (b) If a square matrix A can be written in block form B0 A= 0 C where B and C are smaller square matrices, then 1 is an eigenvalue of A if and only if it is an eigenvalue of either B or C (or both). (c) If A and B are invertible n-by-n matrices, then the products AB and BA have the exact same set of eigenvalues. Give proofs of the above three properties directly from the definition of an eigenvalue.