Find $a,b,c$ so that $\begin{bmatrix} 0 & 1& 0 \\ 0 & 0 & 1\\ a & b & c \end{bmatrix} $ has the characteristic polynomial $-\lambda^3+4\lambda^2+5\lambda+6=0$
Answer
Lets compute the characteristic equation
\[0=\det \begin{bmatrix} -\lambda & 1& 0 \\ 0& -\lambda & 1 \\ a & b & c-\lambda \end{bmatrix} \]
\[=-\lambda \det \begin{bmatrix} -\lambda & 1 \\ b & c-\lambda \end{bmatrix}-1 \det \begin{bmatrix} 0 & 1 \\ a & c-\lambda \end{bmatrix} \]
\[=-\lambda [\lambda (\lambda-c)-b]-[-a]=-\lambda^3+c\lambda^2+b\lambda+a\]
\[=-\lambda^3+4\lambda^2+5\lambda+6.\]
Hence
\[a=6, b=5, c=4.\]
![Daniel90](https://matchmaticians.com/storage/user/100012/thumb/matchmaticians-c8zjsb-file-4-avatar-512.jpg)
443
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 938 views
- $3.00
Related Questions
- Linear independence of functions
- Donald is 6 years older than Sophia. In 4 years the sum of their ages will be 74. How old is Donald now?
- Find the values of x
- Linear Algebra - Matrices (Multiple Choice Question) (1st Year College)
- Find the coordinates of the point $(1,1,1)$ in Spherical coordinates
- Calculate the inverse of a triangular matrix
- Construction Estimate
- Points of intersection between a vertical and horizontal parabola
You should offer some more for this question.