Scientific Computing Using Python - PHYS:4905
Homework #10
Due 10/3/2019


Name _______________________________________________

The answers to these questions must be hand written and work must be shown.


1. (5) Show that inverse of the transpose is the transpose of the inverse,  $(M^{-1})^T = (M^T)^{-1}$.  You may want to use what you know about the product of transposes.


2. (15) Find the inverse of the matrix $$\begin{pmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{pmatrix}$$
When is this matrix singular?


3. (18) Let $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $N = \begin{pmatrix} x & y \\ z & w \end{pmatrix}$. Compute the following:

(a) det(M)
(b) det(N)
(c) MN
(d) det(MN)
(e) The product of your answers to (a) and (b).
(f) How do (d) and (e) compare?


4. (15) Using Cramer's rule, find the inverse of $\begin{pmatrix} 2 & 0 & 1 \\ 7 & 0 & 2 \\ 6 & 2 & 6 \end{pmatrix}$.  Write out the adjoint matrix before evaluating the determinants and then again after.  Check your answer.