1. Prove that if two rows of a matrix are equal, the determinant is zero. Hint: Exchanging the two equal rows causes the sign to change. 2. Using the fact that  show that if  is invertible, then −1  ...

1. Prove that if two rows of a matrix are equal, the determinant is zero. Hint: Exchanging the two equal rows causes the sign to change.

2. Using the fact that
 show that if
 is invertible, then

⁻¹

3. Show that if

^T
and
 is odd, then
 Hint:
ⁿ

 Show the statement is true for
 This is the base case.

 Assume the statement is true for any
 and show it is true for
 Hint: Expansion by minors requires evaluating the determinants of
 matrices.