Pettofrezzo text: 2.1 - 2, 4, 6. 2.2 - 2, 3, 4, 6, 8,12, 16, 17. 2.4 - 2, 4, 7 (explain why), 10. 2.5 - 2, 6.
0. Solve the following equation for the 3x3 matrix A, given C = 3 4 1 and D = 5 6 2
9 2 7 9 4 7
4 1 5 2 1 1
CAD = C-D
1. Find all values of x such that the determinant of each of the matrices below are zero.
a. x-2 1 b. x-4 0 0
-5 x+4 0 x 2
0 3 x-1
2. Prove that the matrices A = (a b) and B = (d e) commute if and only if the determinant of (b a-c) equals zero.
(0 c) (0 f) (e d-f )
3. Using the definition of a determinant we did on class based on permutations, describe how many products are in the determinant of a 4x4 matrix. Why does this imply that the "sweeping diagonal" method does not generalize from the 2x2 and 3x3 cases to the 4x4 case?
4. Using the basic theorems about determinants, determine by hand the determinants of U, U-1, U2 where U = 1 2 3
0 4 5
0 0 6
5. Prove that if A is non-singular (has an inverse), then any power of A is non-singular.
6. For each of the following matrices, decompose it into a product of elementary matrices. Then show the transformation geometrically in stages on the unit square (0,0) (0,1) (1,0) (1,1). Verify that the final picture is the same as doing the transformation in one step.
1 2 -1 2
3 4 2 -1
7. Given real numbers a through d, the set of equations ax+by =kx, and cx+dy = ky, has at least one solution.
a b
c d
a. What is the solution (x,y) when there is exactly one solution?
When there are an infinite number of solutions, then k is called an eigenvalue of the matrix, and the solutions (x,y) are called eigenvectors. An eigenvector is a representative solution for (x,y) that solves the equations for a particular eigenvalue. If (x,y) is an eigenvector then so is any multiple of (x,y).
b. Using the matrix 2 3, calculate all its eigenvalues.
1 5
c. Calculate an eigenvector for each eigenvalue of the matrix 2 3
1 5.
d. Find a 2x2 matrix that has only one distinct eigenvalue.