Math 4 Sample Final 1. 1 0 1 A = 0 2 0 1 0 1 Find the eigenvalues and corresponding eigenvectors for A. Find the orthogonal matrix Q that is used to diagonalize A. 1 2. A= 1 2 2 1 Compute: A1 eigenvalues of A eigenvalues of A1 eigenvalues of A2 eigenvalues of A7 2 2. 1 2 0 A = 1 3 2 0 1 3 Calculate: |A| A1 3 3. A= 1 2 1 0 1 1 1 2 1 1 1 and x = 0 y = 2 B= 0 1 1 1 0 Compute: Rank(B) the eigenvalues of AB trace(BA) the dimensions of a matrix C, if the product CA has dimensions 6 3 ||2x y|| (AB)1 BA xT BAx 4 4. 1 1 1 1 2 1 1 0 A= 1 0 1 2 1 1 1 1 Compute |A|. 5 5. Solve the system of equations by any method. 2x1 x2 + x3 = 7 x1 2x3 + x4 = 7 3x2 x3 + 2x4 = 5 x1 + 2x2 x4 = 1 6 6. 1 0 3 u = 1 v = 1 w = 1 2 1 1 Show that u, v and w are linearly independent. Which pairs are orthogonal? 1 7. Express x = 2 in terms of u, v and w from above. 6 i.e. Find a, b, and c so that au + bv + cw = x 7 8. Draw three level sets of f (x1 , x2 ) = 3x1 x2 Draw three level sets of f (x1 , x2 ) = x2 + x2 1 2 Let P (L, K) = 2L1/3 K 2/3 . Find the MRTS at the point (27, 8). Find the total differential, dP at (27, 8) if dL = .3 and dK = .4. 8 9. Let f (x1 , x2 ) = x3 12x1 x2 + 8x3 . Find stationary points and classify them 1 2 as minimums, maximums or saddle points. 10. Let f (x1 , x2 , x3 ) = 3x2 + 2x2 + x2 x1 + 4x3 . Find stationary points and 1 2 3 classify them as minimums or maximums. 9 11. Let f (x1 , x2 ) = x2 2x1 x2 + 4x2 + 1. Find the maximum and minimum 1 2 values of f when 0 x1 2 and 0 x2 2. 12. Find the minimum value of f (x1 , x2 ) = x2 + x2 subject to the constraint 1 2 x1 x2 = 1. 10 13. The Cobb-Douglas production function for a product is given by P (L, K) = 9L2/3 K 1/3 where L is the number of units of labor and K is the number of units of capital. Each unit of labor costs $40 while each unit of capital costs $120. If $360, 000 is budgeted for the production of this product, what are the values of L and K that maximize production? 14. The technology matrix for an economy based on energy (E) and transportation (T ) is A= .25 .25 .4 .2 Find the output for each sector that is needed to satisfy a nal demand of $50 million for energy and $50 million for transportation. 11 15. A= 2 0 0 3 Find the eigenvalues and eigenvectors of A. 1 0 0 A= 0 0 0 0 0 1 Find the eigenvalues and eigenvectors of A. 12