Spring 97 Exam 2
(1) True or False
- If A is a square matrix, then A^T A = A A^T
- If a 2 x 2 matrix A has 0 as an eigenvalue, then for some
k the matrix A^k is the zero matrix (all entries 0)
- There is a 2 x 2 orthogonal matrix A whose trace is 1.2
- If the columns of a square matrix A are unit vectors, then
det A is less than or equal to 1.
- Let A_n be an n x n matrix with a -1 in the upper right corner,
-1's on the diagonal just below the main diagonal, and 0's
elsewhere. Then det(A_n) = -1 for n = 2,3,4,...
(2) Let A_n be the n x n matrix with 1's everywhere on or below
the main diagonal, and -1's above the main diagonal. Find a
formula for det(A_n) in terms of n.
(3) Let A =
3 0 -.5
0 3 -.5
0 0 .5
-
Find the eigenvalues of A and their algebraic multiplicities.
- Find an eigenbasis for A.
-
Find closed formulas for the entries of A^t w,
where w is the vector 3 e_1 + 4 e_2 + 5 e_3.
(4) Let V be the image of the matrix
1 2
0 -1
0 -2
-2 -4
Find the matrix M of the orthogonal projection from R^4 to
R^4 whose kernel is V.
(5) Let L(x) = Ax be a linear transformation from R^n to R^n
which preserves right angles, i.e. Lx is perp. to Ly if
x and y are perp.
- Show that the columns of A are othogonal
to one another.
- Compute (e_i + e_j) dot (e_i - e_j).
- What is the relationship between the lengths of the columns
of A?
- Show that L is an othogonal transformation followed
by a dilation.