Exercise 3. Determinants

Experiment with the appropriate matrices to find answers to the following questions. Show me the matrices you used.

a. In your own words, how do you compute the determinant of a diagonal matrix?

b. How do you compute the determinant of an upper or lower triangular matrix?

c. Consider matrices with a row of zeroes. What happens to their determinants?

d. Is Det[A+B]=Det[A]+Det[B]?

e. Is Det[AB]=Det[A]Det[B]?

f. What is the relationship between Det[A] and Det[Inverse[A]]?

g. Use part f. to give a condition involving the determinant for a matrix to be nonsingular.

h. How does Det[A] relate to Det[Transpose[A]]?

i. Use part h. (do not use Mathematica) to show that Det[A.Transpose[A]]>0.