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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.$$\left[\begin{array}{rrr}{-1} & {0} & {1} \\ {-3} & {4} & {1} \\ {0} & {0} & {2}\end{array}\right]$$

$\operatorname{det}(A-\lambda I)=\operatorname{det}\left[\begin{array}{ccc}{1-\lambda} & {0} & {1} \\ {-3} & {4-\lambda} & {1} \\ {0} & {0} & {2-\lambda}\end{array}\right]=(2-\lambda) \operatorname{det}\left[\begin{array}{cc}{1-\lambda} & {0} \\ {-3} & {4-\lambda}\end{array}\right] =(2-\lambda)(-1-\lambda)(4-\lambda)=-\lambda^{3}+5 \lambda^{2}-2 \lambda-8$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 2

The Characteristic Equation

Vectors

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Okay, so we have a minus the eye, which is one minus lambda. 1234 Minus Lambda 1002 Money's Lambda. Now let's take a determinate of that. All right, That's difficult to to minus Lambda everyone one in London for money slammed it. Excuse me on the Cube. Negative barbecue with post its lander squared minus to London Markazi.

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