5 Linear Algebra II

1. Given the matrices
   
   $$A = \left( \begin{array}{cc} -2 & 2 0 & 1 14 & 2 \... ...cc} 3 & 4 2 & 1 14 & 16 1 & 25 \end{array} \right)$$
   
   
   find, if they exist,
   • $-5 A + 3B$;
   • $A^{\rm T}$;
   • $AB$, $BA$, $A^{\rm T}B$, and $BA^{\rm T}$;
   • the unit matrix or unit matrices that $A$ can be pre-multiplied by, demonstrating that this does not change $A$;
   • the unit matrix or unit matrices that $A$ can be post-multiplied by, demonstrating that this does not change $A$;
   • the zero matrix or matrices that can be added to $A$, stating what the result will be;
   • the zero matrix or matrices that $A$ can be pre-multiplied by, stating what the result will be;
   • the zero matrix or matrices that $A$ can be post-multiplied by, stating what the result will be.
   Double-check you are using the correct matrices while doing all the above. Make sure you have answered everything. [2, chapter 7].

2. Consider the system of equations
   $$\begin{eqnarray*} \phantom{-}4 x_1 - \phantom{0}x_2 + 4 x_3 & = & 1 \\ \phant... ...2 - 5 x_3 & = & 0 \\ - 2 x_1 + \phantom{0}x_2 + 7 x_3 & = & 4 \end{eqnarray*}$$
   
   
   Solve this system, as written, using the class procedure. In the first stage of the forward elimination, you will first need to do a partial pivoting to avoid fractions. In the second stage, partial pivoting does no good, so should not be done. Since you cannot take an integer multiple of the original equation greater than 1 in this stage (since it would mess up the $L$ matrix asked later) you will need to live with fractions in the final equation. Solve the resulting equations using the class procedure. Next do the same but using augmented matrix notation. Next find the $L$ and $U$ matrices of the $LU$ decomposition and multiply the result. Do they give back the original matrix $A$ of the system. If not, what do they give back? So, what is the right hand vector $\vec b^{\rm pp}$ to use when solving $L\vec y=\vec b^{\rm pp}$? Solve this system and check that you get indeed the correct right hand side vector to use in solving the system $U\vec x=\vec y$. Double-check you are using the correct matrix and right hand side while doing all the above. Make sure you have answered everything. [2, chapter 7].