Science:Math Exam Resources/Courses/MATH152/April 2012/Question 08 (a)

MATH152 April 2012

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Question 08 (a)
Consider the 2 x 2 system of differential equations $\mathbf {x} '(t)={\begin{bmatrix}-1&-2\\2&-1\end{bmatrix}}\mathbf {x} (t)$ where $\mathbf {x} (t)={\begin{bmatrix}x_{1}(t)\\x_{2}(t)\end{bmatrix}}$ is the vector of unknown functions. Find all eigenvalues that may or may not be complex valued.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Find the eigenvalues of the matrix $A={\begin{pmatrix}-1&-2\\2&-1\end{pmatrix}}$.

Hint 2
The eigenvalues of $\displaystyle A$ are calculated by ${\text{det}}(A-\lambda I)={\text{det}}{\begin{pmatrix}-1-\lambda &-2\\2&-1-\lambda \end{pmatrix}}$ and find $\displaystyle \lambda$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. To calculate the eigenvalues of the matrix $A={\begin{pmatrix}-1&-2\\2&-1\end{pmatrix}}$, we find $\displaystyle \lambda$ such that ${\begin{aligned}0&={\text{det}}{\begin{pmatrix}-1-\lambda &-2\\2&-1-\lambda \end{pmatrix}}\\&=(-1-\lambda )^{2}+4\\&=\lambda ^{2}+2\lambda +5\end{aligned}}$ To calculate $\displaystyle \lambda$, we use the quadratic formula, $\lambda _{1,2}={\frac {-2\pm {\sqrt {4-20}}}{2}}=-1\pm 2i$ where $\displaystyle {\sqrt {-1}}=i$. Hence, the eigenvalues $\displaystyle \lambda _{1,2}$ are $\displaystyle -1+2i$ and $\displaystyle -1-2i$.