Question

please choose your favorite, unique 3x3 Matrix A containing no more than two 0 entries and having a nonzero determinant. I suggest choosing a matrix with integer elements (e.g. not fractions or irrational numbers) for computational reasons.

1. What is your matrix A? What is det (A)?
2. What is A^T? What is det (A^T)?
3. Calculate A A^T. Show that A A^T is symmetrical.
4. Calculate A^T A
5. Calculate the determinant of (A A^T) and the determinant of (A^T A). Should the determinants be equal? Why or why not?
6. Find the inverse of A using the method: [A | I ] → [ I | A^-1 ]. Set up and then use a calculator (recommended). Express the elements of A^-1 as fractions if they are not already integers. (Use Math -> Frac if needed.)
7. Begin the LU factorization of A by determining a first elementary matrix E1 and its inverse E1^-1. Identify the associated row operation. (That is all. You do not need to complete the LU factorization for the exam.)

Answer 1