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$[\mathbf{M}]$ Let $A=\left[\begin{array}{rrr}{-6} & {28} & {21} \\ {4} & {-15} & {-12} \\ {-8} & {a} & {25}\end{array}\right] .$ For each value of $a$ in the set $\{32,31.9,31.8,32.1,32.2\},$ compute the characteristic polynomial of $A$ and the eigenvalues. In each case, create a graph of the characteristic polynomial $p(t)=\operatorname{det}(A-t I)$ for $0 \leq t \leq 3 .$ If possible, construct all graphs on one coordinate system. Describe how the graphs reveal the changes in the eigenvalues as $a$ changes.

The characteristic polynomials and the eigenvalues for the various values of a are given in the following table:

$\begin{array}{|c|c|c|}\hline a & {\text { Characteristic Polynomial }} & {\text { Eigenvalues }} \\ \hline 31.8 & {-4-2.6 t+4 t^{2}-t^{3}} & {3.1279,1,-.1279} \\ \hline 31.9 & {.8-3.8 t+4 t^{2}-t^{3}} & {2.7042,1,2958} \\ \hline 32.0 & {2-5 t+4 t^{2}-t^{3}} & {2,1,1} \\ \hline 32.1 & {3.2-6.2 t+4 t^{2}-t^{3}} & {1.5 \pm .97477 i, 1} \\ \hline 32.2 & {4.4-7.4 t+4 t^{2}-t^{3}} & {1.5 \pm 1.4663 i, 1} \\ \hline\end{array}$

The graphs of the characteristic polynomials are:(See graph)

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 2

The Characteristic Equation

Vectors

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So here we have given metric saying the question and we have to find a characteristic polynomial off A and Eigen values. So we and they were also given the various values off the week and youth. So first thing that if we were to find characteristic paranormal valued and we need a one equation and the question is, they're too determined. Mind nambla All right equals two fetal. Before taking this equation, we have to consider value of faith. And the various metals are given in the question, though among all of them. Here I take eight equals to 32 for now. Over metrics. He my next six for minus eight 28 minus 15 turkey do one minus doing 25 through. After taking it, we hope to find a political equation and a plane Tector equation. Into this metric, we can get a equals two minus six miners glamorous for minus a 28 minus 15. My most Lamela 30 do 21 minus will 25 minus lemon. I think fool. Before this determinant, we need another equation. And that special question is like this limra u minus. Yes, one lember squared. Plus is Lemba minus is three. 30 equals 20 So here it's one is the some off the principal Bagnall elements. And by doing that, we can get minus food. It s too ID equals toe some off my north off the principal off the determinant. By following that, we can get five and s three. Finally, it is different form off our metrics into determinant and for living over metric determinant. You can get the value off history and thirties minor by hoping 23 values into the equation we can get were well, you know, really question. And that is minus Limerick. You plus four liberal Square minus five lambada both to delete the final equation. You know, Mary, question for value for eight equals 32. And the next thing that we have to find is the I got venues for the same value off a vehicle to 32 go find. Hi. Good. Then youth for 18 holes to 32. Then you re open full a boat. Question this excavation with the help offer triangular through and by following 30 question of triangular through became irrigate minus LaMotta, my next cricket one into another record minus Lamela minus one that is plus and again minus into Lebanon. Minor. So finally, question is minus Lemba, y know, swollen to record. Lamoreaux minus one record close. Break it, emerald miners to break it close. So by that we can get the three various to various values off the limb brand that is 12 So we can easily right that Eigen values in this form and the form is 11 We can write it. Anyone effort even convicted. That's it. And full by taking another value off day by teeth maker, we can find the other polynomial characteristic polynomial equation as well, if I got released.

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