Math 4 Sample Midterm 1. \u0014 A= 1 1 1 1 \u0015 Find the eigenvalues and corresponding eigenvectors for A. Find the orthogonal matrix Q that is used to diagonalize A. Find and linear mapping. 1 A. Make a sketch of A as a 2. 3 1 1 A = 1 1 1 1 0 1 Calculate: trace(A) rank(A) |A| adj(A) A1 2 3. \u0014 A= 1 1 2 0 1 1 \u0015 1 1 1 2 B = 2 2 x = 2 and y = 1 1 1 1 0 Compute: Rank(B) AB xyT xT y ||2x + y|| 3 4. Let A be a square matrix such that AAT = I. Show that |A| = 1. 5. Prove that |AB| = |BA|. 6. 4 2 1 1 3 1 1 2 A= 1 2 1 3 2 1 1 3 Compute |A|. 4 7. Solve for x3 the system of equations by Cramer's rule. 7x1 8x2 + 2x3 = 3 4x1 5x2 + x3 = 9 2x1 3x2 2x3 = 5 8. Suppose |A| = 2, A while |B| = 3. A and B are 3 3 matrices. Compute the following. |AB| | 3A| |B1 | |A3 | |(AT )(2I)| 5 9. Solve the system of equations by any method. 2x1 x2 + x3 = 3 x1 2x3 + x4 = 3 3x2 x3 + 2x4 = 11 x1 + 2x2 x4 = 1 10. Solve the following underdetermined system of equations. 2x + y + z = 2 x+yz =1 6 11. 3 1 0 1 0 u= v= w= 1 1 3 2 Show that u, v and w are linearly independent. Which pairs of vectors are orthogonal? 3 Express x = 5 in terms of u, v and w. 7 i.e. Find a, b, and c so that au + bv + cw = x 7 12. The technology matrix for an economy based on Steel (S) and Lumber (L) is \u0014 \u0015 .3 .25 M= .1 .25 Find the output for each sector that is needed to satisfy a final demand of $40 million for steel and $40 million for lumber. 13. Solve the following equation for X. AX + X = B where \u0014 A= 1 1 1 0 \u0015 \u0014 B= 8 0 1 1 0 \u0015