[SOLVED] MATH307-Group Homework 10

Description

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 1

multiple of one row to another row elementary row operation matrix), D =

Ñ2 0 0é Ñ 1 4 −1é

• −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(B^T)²|,|(B^T)⁻¹A³|.
4. Suppose that Q is a n × n real orthonormal matrix, i.e., QQ^T = I. Find the possible values for |Q|.