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This is Exercise 4.3.24 in Linear algebra by Friedberg.

Let $A \in M_{n \times n} $ have the form $\begin{pmatrix} 0 & 0 & 0 & ... & 0 & a_0\\ -1 & 0 & 0 & ... & 0 &a_1 \\ 0 & -1 & 0 & ... &0 & a_2 \\\vdots &\vdots &\vdots &...&\vdots&\vdots \\ 0 & 0 & 0 & . . . & -1 & a_{n-1} \end{pmatrix}$

Copmpute $\det(A + tI)$, where $I$ is the $n \times n$ identity matrix.

I was thinking about expressing $A$ into the product of two matrices and using the property that $\det(A)$ is the product of the determinant of each matrix. But, I am stuck here, and I am not sure if this is the right way to solve this question. I appreciate if you give any help.

shk910
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    Don't go into this complicated way (product of matrices). You are (up to minus signs) asked to compute the characteristic polynomial of what is known as a "companion matrix". – Jean Marie Aug 13 '19 at 06:53
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    Perhaps you can induct on $n$ and use cofactor expansion to compute the determinant. – Kenneth Yeo Aug 13 '19 at 06:57
  • See https://math.solverer.com/library/stephen_friedberg/linear_algebra/exercise_4-3-24?utm_source=stackexchange.com&utm_medium=link – togrultopal Dec 08 '21 at 11:05

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