Description
1. Find the determinants of the following matrices: A =
Ñ
0 1 0
1 0 0
0 0 1é
(a permutation elementary row operation matrix), B =
Ñ
1 0 0
0 3 0
0 0 1é
(a multiplication elementary row operation matrix), C =
Ñ
1 0 0
−1 1 0
0 0 1é
(an adding a
multiple of one row to another row elementary row operation matrix),
Ñ
D =
2 0 0
1 −5 0
0 0 3é
, E =
Ñ
1 4 −1
−1 1 0
2 0 1 é
.
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|.
