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Exercises $9-14$ require techniques from Section $3.1 .$ Find the characteristic polynomial of each matrix, using either a cofactor expansion or the special formula for $3 \times 3$ determinants described prior to Exercises $15-18$ in Section $3.1 .$ INote: Finding the characteristic polynomial of a $3 \times 3$ matrix is not easy to do with just row operations, because the variable $\lambda$ is involved.$$\left[\begin{array}{rrr}{6} & {-2} & {0} \\ {-2} & {9} & {0} \\ {5} & {8} & {3}\end{array}\right]$$

$-\lambda^{3}+18 \lambda^{2}-95 \lambda+150$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 2

The Characteristic Equation

Vectors

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so big. I matrix six negative, too. Zero negative too. Nine, zero, five, eight and three. Like that. So once again, we're going to subtract lambda from all of the diagonal diagonal entries and use the equation that we used previously to try and find the characteristic polynomial. So the determinants of this matrix is equal to six minus lambda times the determinant. Ah, nine minutes, Lambda zero eight three minutes. Lambda minus native to times the determinant of negative too. Zero five three minus lander. And then plus zero times the determinant. Uh, negative, too. Nine minutes, Landau. Five and eight. Now we can cancel out the last term because with the zero and the coefficient is gonna end up being zero. Anyway, it's and we're left with these and we're gonna expand them out like this. Six minutes. Land a times, part disease, nine minutes, lamb that times three minus lander minus zero times eight. I know that's gonna cancel out into zero, but let's leave that, as is for now, just to keep the form and then plus two times currencies negative. Two times three months, Lambda minus zero times mine like that. And then we mark the zeros outs like that, and then we're left with six months. Lander times nine minus Lambda Times three minus lander plus negative four times three minus lander. So now we're going to expand these out so that we can get the polynomial. If we were trying to get the roots, we would be trying to fact arises by, for example, group into three months. Land us together and then expanding the polynomial created by six minutes. Linda Times Nyman's Lambda minus four. But this is asking for the polynomial. Instead of the Eigen values, we're going to expand them. Little left term becomes negative Land a cube plus e team Lambda squared minus minus 99 lander and then finally plus 162. And then the expression on the right expands to for Orlando minus 12 making this in total minus land. Acute plus 18 land a squared minus 95 Lambda plus 1 50. And we're done

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