Ñ0 1 0é
1. Find the determinants of the following matrices: A = 1 0 0 (a per-
0 0 1
Ñ1 0 0é
mutation elementary row operation matrix), B = 0 3 0 (a multipli-
0 0 1 Ñ 1 0 0é
cation elementary row operation matrix), C = −1 1 0 (an adding a
0 0 1
multiple of one row to another row elementary row operation matrix), D =
Ñ2 0 0é Ñ 1 4 −1é
1 −5 0 ,E = −1 1 0 .
0 0 3 2 0 1
2. Let A be an invertible matrix, one can prove that |A| 6= 0, find the determinant of A−1 in terms of |A|.
3. If |A| = 2,|B| = −1, find |A−1(BT)2|,|(BT)−1A3|.
4. Suppose that Q is a n × n real orthonormal matrix, i.e., QQT = I. Find the possible values for |Q|.
