Linear Algebra with Applications, 10th Edition

Section 6.6 Exercises

1. Find the matrix associated with each of the following quadratic forms:

   1. $3x^2-5xy+y^2$

   2. $2x^2+3y^2+z^2+xy-2xz+3yz$

   3. $x^2+2y^2+z^2+xy-2xz+3yz$

2. Reorder the eigenvalues in Example 2 so that ${\lambda }_1=4$ and ${\lambda }_2=2$ and rework the example. In what quadrants will the positive x′ and y′ axes lie? Sketch the graph and compare it to Figure 6.6.3.

3. In each of the following, (i) find a suitable change of coordinates (i.e., a rotation and/or a translation) so that the resulting conic section is in standard form, (ii) identify the curve, and (iii) sketch the graph:

   1. $x^2+xy+y^2-6=0$

   2. $3x^2+8xy+3y^2+28=0$

   3. $-3x^2+6xy+5y^2-24=0$

   4. $x^2+6xy+y^2+3x+y-1=0$

4. Let ${\lambda }_1$ and ${\lambda }_2$ be the eigenvalues of

   $$A=\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]$$

   What kind of conic section will the equation

   $$a{x}^2+2bxy+c{y}^2=1$$

   represent if ...