- Solve $ \det\left[ \begin{array}\\ 2-\lambda & 4 \\ 3 & 1-\lambda \end{array} \right] = 0$. The values of $\lambda$ that satisfy this equation are called the eigenvalues of the matrix $ \left[ \begin{array}\\ 2 & 4 \\ 3 & 1 \end{array} \right] $.
- Repeat problem 1 for the matrix $ \left[ \begin{array}\\ 8 & 3 \\ 7 & 12 \end{array} \right] $.
- Given $f(x,y)=x^2-y^2-2x+4y+6$
- Find critical points of $f$ (where $\nabla f = \vec{0}$).
- Compute the second derivative $D^2f$ (should be a 2x2 matrix)
- At each critical point, evaluate the second derivative, then find the eigenvalues of the resulting matrix (of numbers).
- Determine whether each critical point is a local maximum, local minimum, or saddle point.
- Suppose $\vec{u}$ and $\vec{v}$ are given.
- If $\vec{u}$ and $\vec{v}$ were parallel, how would you express this fact as a vector equation?
- Are the vectors $\left< 2,8 \right>$ and $\left<-1,-4 \right>$ parallel?
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