5 Linear Algebra II

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

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, and demonstrating that this does not change $A$;
   • the unit matrix or unit matrices that $A$ can be post-multiplied by, and 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, 8.1].

2. Consider the system of equations
   $$\begin{eqnarray*} \phantom{-}\phantom{0}x_1 + \phantom{0}x_2 - 5 x_3 & = & 0 \\... ...2 + 4 x_3 & = & 1 \\ - 2 x_1 + \phantom{0}x_2 + 7 x_3 & = & 4 \end{eqnarray*}$$
   
   
   Solve this system, as written (not in matrix notation), using the class procedure

   Since you cannot take an integer multiple of the original equation different from 1 in this question (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 exactly the same but using augmented matrix notation. (See also the revised notes on linear algebra.)

   Next find the $L$ and $U$ matrices of the $LU$ decomposition and multiply the result. Verify that $LU$ gives back the original matrix $A$ of the system. Verify that solving $L\vec b^*=\vec b$ and then $U\vec x=\vec b^*$ gives the correct solution to the system. Double-check you are using the correct matrix and right hand side while doing all the above. Make sure you have answered everything. [2, 8.1].