1. (2 pt) Here is an identity for the exponential function er. in n ! We can define the exponential of a matrix in a similar fashion: exp (A) :- An n ! Suppose that A had eigenvector V with eigenvalue ). Prove that V is an eigenvector of exp(A) with eigenvalue el. Hint: Recall that if V is an eigenvector of A with eigenvalue ), then V is an eigenvector of An with eigenvalue 1", for all n 2 0. 2. (2 pt) Let A be a diagonal matrix. Prove that exp(A) is diagonal, and the elements are the exponents of the elements of A, that is, (exp(A) )ij = eij. Hint: What property can we say about (A");, if A is diagonal? 3. (7 pt) We say that A is an isometry if it preserves Eucledian distance, e.g. for all Let A - c d be an arbitrary 2 x 2 matrix. Find the set of all A E R2x2 that satisfy both following properties: . A is an isometry. . All entries along the main diagonal are strictly positive. 4. (2 pt) Hence, or otherwise, find an isometic matrix such that the top left entry of the matrix is neither zero nor one