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Q5.2-13E

Expert-verifiedFound in: Page 267

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

**Question: Exercises 9-14 require techniques 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 Exercise 15-18 in Section 3.1. [Note: 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.**

** **

**13. \(\left[ {\begin{array}{*{20}{c}}6&- 2&0\\- 2&9&0\\5&8&3\end{array}} \right]\)**

The characteristic polynomial of the matrix is \( - {\lambda ^3} + 18{\lambda ^2} - 95\lambda + 150\).

The **eigenvalue** of an \(n \times n\) matrix \(A\) is a scalar \(\lambda \) such that \(\lambda \) satisfies the characteristic equation \(\det \left( {A - \lambda I} \right) = 0\).

When \(A\) is an \(n \times n\) matrix, \(\det \left( {A - \lambda I} \right)\) is the **characteristic polynomial of** \(A\), which is the polynomial of degree \(n\).

Use the cofactor expression down the third column to obtain the characteristic polynomial of the matrix, as shown below.

\[\begin{array}\det \left( {A - \lambda I} \right) = \det \left[ {\begin{array}{*{20}{c}}{6 - \lambda }&{ - 2}&0\\{ - 2}&{9 - \lambda }&0\\5&8&{3 - \lambda }\end{array}} \right]\\ = \left( {3 - \lambda } \right)\det \left[ {\begin{array}{*{20}{c}}{6 - \lambda }&{ - 2}\\{ - 2}&{9 - \lambda }\end{array}} \right]\\ = \left( {3 - \lambda } \right)\left[ {\left( {6 - \lambda } \right)\left( {9 - \lambda } \right) - \left( { - 2} \right)\left( { - 2} \right)} \right]\\ = \left( {3 - \lambda } \right)\left( {{\lambda ^2} + - 15\lambda + 50} \right)\\ = - {\lambda ^3} + 18{\lambda ^2} - 95\lambda + 150\\ = \left( {3 - \lambda } \right)\left( {\lambda - 5} \right)\left( {\lambda - 10} \right)\end{array}\]

Thus, the characteristic polynomial of the matrix is \( - {\lambda ^3} + 18{\lambda ^2} - 95\lambda + 150\).

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