Consider the following 2x2 matrix A: [2 3 3 2] Answer the following questions. You should show your work for all these. Computing the answers using calculators or MATLAB is NOT allowed. 1. Is A symmetric? (a matrix is symmetric if the transpose of the matrix is equal to itself) 2. What is the determinant of A? 3. What is the inverse of A? 4. What are the eigen values of A? 5. What are the eigen vectors of A? (Assume that eigen vectors are normalized such that they are "unit vectors"). Which is the principal eigen vector? (The principal eigen vector is the one that corresponds to the principal eigen value) 6. Compute the dot product of the eigen vectors of A. What does the answer tell you? (optional: Do you know if this answer will be the same whatever the initial matrix A?) 7. Let one of the eigen vectors you computed in part 5 be v, and the correspondng eigen value l. Compute AXv (where X is the matrix multiplication of A with v). Show that AXv is actually l*v (that is multiplying the eigen vector with A only increases its size). 8. Consider the random column vector v=[1 4] Compute AXv ; AXAXv; AXAXAXv. Show that these vectors are increasingly becoming parallel to the principal eigen vector (part 5). One way of doing this is to compute the unit vectors corresponding to the three vectors AXv, AXAXv and AXAXAXv and show that they are becoming closer and closer to the principal eigen vector (in terms of the coefficients). It is also illustrative to draw the unit vectors on a 2-D plot and see how they move towards the principal eigen vector. [Notice that part 8 is essentially telling you an alternate way of computing the principal eigen vector of a given matrix. This is called the "POWER" method. You will see that Google's page-rank computation essentially involves a power method identical to this.]