Due: Monday, April 14, 2008.
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, 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
tranformation 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 a through d, the
set of equations ax+by =kx,
and cx+dy = ky, has at least one solution. 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.
a b
c d
8. Using the matrix 2 3,
calculate all its eigenvalues. An eigenvector is a representative
solution(s) for (x,y) that solves the
1 5
equations for a particular eigenvalue.
Calculate the eigenvector(s) for each eigenvalue of the matrix 2 3
1 5.
9. Find a 2x2 matrix that has only one
eigenvalue.