Today, in class, I answered some homework questions from Section 1.4, including the following problems:

- Show that an upper triangular matrix with non-zero diagonal entries is nonsingular. (We’ve not covered determinants yet so we used a row-equivalence to the Identity argument)
- Show that the inverse of an upper triangular matrix is upper triangular. (We used the fact that the same row operations that reduce a matrix to the identity will reduce the identity to the inverse of a matrix. The row operations that reduce an upper triangular matrix to the identity will necessarily change the identity into an upper triangular matrix.)
- Given the matrices, [tex]A[/tex] and [tex]C[/tex], solve the matrix equation: [tex]XA + C = X[/tex].

The last homework example was over the Vandermonde matrix system, proving that the Vandermonde system is equivalent to polynomial interpolation. We also proved that if the values of the [tex]\mathbf{x}[/tex]-vector (independent variables) are distinct, then the Vandermonde matrix is non-singular. As you might imagine, I got very excited and energetic about explaining this example. Polynomial interpolation was one of the first topics from Numerical Analysis that I fell “in love” with. It set me down the path to becoming a numerical analyst.

Originally, we had a text scheduled for the next class but because I spent a significant portion of class time on polynomial interpolation, I decided to postpone the exam until after I can answer some questions from the review and last section’s homework. We’ll do this next time and if there are no questions we’ll move on to the next chapter: a short one on determinants.