EGR 599 ______

LAST NAME, FIRST

Problem set #2

1. (3.7[1]) Find the point of intersection (if any) for the following lines:

a.  x - 2y + 1 = 0

2x + 3y - 7 = 0

b.  x - 2y + 11 = 0

- x + 2y - 13 = 0

2. (3.111) Let A =

Compute the inverse of A in the following way:

a. Compute the determinant matrix B such that bij = det(Aij) and use the cofactors

b. Transpose and divide by det(A)

3. (3.191) Without typing in all the elements, use MATLAB to create a 6´6 matrix with 4’s on the main diagonal and -1 on the first upper diagonal and first lower diagonal. This matrix is called a band matrix with the form

I =

4. (3.231) Using the rank of the matrix whose rows are the vectors

[3 0 2 2];[-6 42 24 54]; [21 - 21 0 -15],

do the following:

a. Determine the number of linearly independent vectors

b. If the set of vectors is dependent, write the dependent vector in terms of the independent ones.

5. (3.271) Write an M-function that rotates the 2D vector x by q degrees. Test the function with vectors (1, 0) and (0, 1) using q = 30o.

6. (P. Chapra 17.6) Use least-squares regression to fit a straight line, y = a + bx, to

x / 2 / 3 / 4 / 7 / 8 / 9 / 5 / 5
y / 9 / 6 / 5 / 10 / 9 / 11 / 2 / 3

(a) Along with the slope and intercept, compute the standard error of the estimate and the correlation coefficient. Plot the data and the straight line. Assess the fit.

(b) Recompute (a), but use polynomial to fit a parabola, y = a + bx + cx2, to the data. Compare the results with those of (a).

7. (3.351) Find the inverse and verify the result for the matrix that relates spherical and rectangular coordinates.

Q =

8. (3.371) Write MATLAB statements to determine if a matrix A is upper triangular or lower triangular. Hint: Investigate the commands any, all, tril, and triu.

9. (3.391)

a. For an n´n matrix A, how are det(2A) and det(-A) related to det(A)?

b. Suppose A, B, and D are 2´2 matrices. Find the determinant of the of the 2´2 matrix formed as

c. If B = M-1 A M, show that det(B) = det(A)

[1] Advanced Engineering Mathematics with MATLAB, 2e by Thomas Harman