Practice Test IV Actual test is Wednesday, April 20, 2005 1. Find an orthonormal basis of the Null Space of the matrix A where | 1 2 1 0 1 | A = | 2 5 1 1 0 | 2. Find the best parabola fit to the points ( -2,3), (-1, 2), (0,1), ( 1, 2), ( 2,3). A = {{1,-2,4}, {1,-1,1}, {1, 0,0}, {1, 1,1}, {1, 2,4}}; B = {3,2,1,2,3}; Inverse[Transpose[A].A].Transpose[A].B 47 3 Out[25]= {--, 0, -} 35 7 47/35 + 0 x + 3/7 x^2; ------------------------------------------------- p1 = ListPlot[{{-2,3},{-1,2},{0,1},{1,2},{2,3}}]; p2 = Plot[3/7 x^2 + 47/35,{x,-2,2}]; p3 = Show[p1,p2]; Display["parabola.ps",p3]; ------------------------------------------------ 3. Write down the solution to the differential Equation | y1 | | 5 0 0 0 0 0 | | y1 | | y2 | | 0 5 1 0 0 0 | | y2 | | y3 | = | 0 0 5 1 0 0 | | y3 | | y4 | | 0 0 0 5 1 0 | | y4 | | y5 | | 0 0 0 0 5 0 | | y5 | | y6 | | 0 0 0 0 0 5 | | y6 | -1 4. Find a matrix P such that P A P is in Jordan canonical form where | -9 -12 -22 -11 | A = | 2 4 4 2 | | 5 6 12 5 | | -3 -4 -6 -1 | Hint: A has 0 and 2 as eigen values. A = {{ -9,-12,-22,-11 }, { 2, 4, 4, 2 }, { 5, 6, 12, 5 }, { -3, -4, -6, -1 }}; eigenvalue 0 1 0 0 -5 0 1 0 1 0 0 1 2 0 0 0 0 eigenvalue 2 1 0 2 1 0 1 0 0 0 0 0 0 0 0 0 0 x -2 -1 y = 0 0 z 1 0 w 0 1 A = {{ -9-2,-12,-22,-11,-2r-s }, { 2, 4-2, 4, 2,0 }, { 5, 6, 12-2, 5,r }, { -3, -4, -6, -1-2,s }}; 1 0 2 1 -r 0 1 0 0 r 0 0 0 0 r + s 0 0 0 0 -r - s To extend, r+s = 0 Let r = 1 and s = -1 P = -1 -1 -1 5 0 1 0 -1 1 0 0 -2 -1 0 1 1 P = {{ -1, -1, -1, 5}, { 0, 1, 0, -1}, { 1, 0, 0, -2}, { -1, 0, 1, 1}}; MatrixForm[Inverse[P].A.P] Check: 2 1 0 0 0 2 0 0 0 0 2 0 0 0 0 0 Tidbits 1. Write down the definition of the determinant. Det[A] = Sum sgn(p) a a ... a all p 1 p(1) 2 p(2) n p(n) | x 5 1 | 2. Solve Det|x+1 6 2 | = 12 | 2x 3 -1 | | x 5 1| | 0 5 1 | Det| x 6 2|+Det| 1 6 2 | = 12 | 2x 3 -1| | 0 3 -1 | | 1 5 1| | 0 5 1 | x Det| 1 6 2|+Det| 1 6 2 | = 12 | 2 3 -1| | 0 3 -1 | | 1 5 1| | 0 5 1 | x Det| 1 6 2|+Det| 1 6 2 | = 12 | 2 3 -1| | 0 3 -1 | x(-6+20+3-(12+6-5) ) + 8 = 12 4x +8 = 12 x = 1 -------------------------------------------------- Check: Det[{{1,5,1},{2,6,2},{ 2,3,-1}}] = 12 Det[{{1,5,1},{1,6,2},{2,3,-1}}] -------------------------------------------------- 3. How do the elementary row operations affect the determinant? (a) Interchange two rows changes the sign (b) Multiply a row by c multiplies the determinant by c. (c) Add a multiple of one row to another does not change the determinant. 4. If Det[A] = 7 and Det[B] = 6, what is Det[A B B A] ? 7 6 6 7 = 1764 5. A is square. What are the possibilities of Det[ RCF[A] ]? 0 or 1 6. What is the formula for expansion of a determinant along the i th column? n k+i Det[A] = SUM (-1) a Det[A ] k = 1 ki ki 7. Write the solution for x using Cramer's rule. Do not expand the determinants. x + y - z + w = k+1 x - y - w = k-1 x + y - z + w = 2k + 1 x - y + z + w = 2k - 1 | k+1 1 -1 1 | Det | k-1 -1 0 -1 | |2k+1 1 -1 1 | |2k-1 -1 1 1 | x = --------------------- | 1 1 -1 1 | Det | 1 -1 0 -1 | | 1 1 -1 1 | | 1 -1 1 1 | 8. What is the relationship between the value of the determinant and the existence of the inverse? The inverse exists if and only if the determinant is not zero. 9. What is the adjoint of a matrix? | +Det[A11] -Det[A21] +Det[A31] -Det[A41] ... | | -Det[A12] +Det[A22] -Det[A32] +Det[A42] ... | | +Det[A13] -Det[A23] +Det[A33] -Det[A43] ... | adj(A) = | -Det[A14] +Det[A24] -Det[A34] +Det[A44] ... | | . . . . | | . . . . | . . . . | | 3 2 1 | 10. Find Adj | 1 0 1 | | 0 1 2 | check | 3 2 1 | |-1 (2) 1 | T | -1 -3 2 | | -6 0 0 | Adj | 1 0 1 | = |(3) 6 (3) | = | -2 6 -2 | | 0 -6 0 | | 0 1 2 | | 2 (2) -2 | | 1 -3 -2 | | 0 0 -6 |