MATH 54 QUIZ 3-
1. Compute the determinant of the following matrix.
2. Give an example of a 4 × 4 matrix A such that the 16 coefficients of A are pair-wise distinct and det A = 0.
3. Let
It is given that A is row equivalent to B. Find bases for Col A and Nul A.
4. Which ordered pairs of integers (m, n) can be of the form (dim Col A, dim Nul A), where A is a 5 × 8 matrix?
5. Let S be the parallelogram determined by the vectors
Compute the area of the image of S under the mapping x |→ Ax.
6. Let A, B be n × n matrices with integer coefficients. Suppose that det(AB) = 3. Prove that A, B are invertible, and that one of A-1 and B-1 have integer coefficients.
7. Write anything you like here (comments, questions, suggestions, etc).