Please provide a justification to your responses to these questions.

  1. In this section, our hypothetical computer could only store numbers using 3 decimal places. Most computers can store numbers using 15 or more decimal places. Why do we still need to be concerned about the accuracy of our computations when solving systems of linear equations?

  2. Finding the \(LU\) factorization of a matrix \(A\) is roughly the same amount of work as finding its reduced row echelon form. Why is the \(LU\) factorization useful then?

  3. How can we detect whether a matrix is invertible from its \(LU\) factorization?