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Title: Characteristic Polynomial
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Series: Linear Algebra
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Chapter: Eigenvalues and similar things
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YouTube-Title: Linear Algebra 54 | Characteristic Polynomial
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Bright video: https://youtu.be/Pt_p_BOsAsA
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Dark video: https://youtu.be/CgmJeDZSLMY
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: la54_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $A \in \mathbb{R}^{n \times n}$ be a square matrix. What is the correct definition for the characteristic polynomial?
A1: $\lambda \mapsto \det(A - \lambda \mathbf{1} )$
A2: $\lambda \mapsto \det(A + \lambda \mathbf{1} )$
A3: $\lambda \mapsto \det( \lambda A - \mathbf{1} )$
A4: $\lambda \mapsto \det( \lambda A + \mathbf{1} )$
Q2: Let $A \in \mathbb{R}^{n \times n}$ be a square matrix. What is not correct for the characteristic polynomial $p_A(\lambda) = \det(A - \lambda \mathbf{1} )$?
A1: The leading coefficient of $p_A$ is always 1.
A2: $p_A$ is a polynomial of degree $n$
A3: $p_A(0) = \det(A)$
A4: $p_A(\lambda) = (-1)^n \lambda^n + \cdots + c_1 \lambda + c_0 $
Q3: Let $A \in \mathbb{R}^{2 \times 2}$ be a square matrix given by $\begin{pmatrix} 2 & 1 \ 1 & 3 \end{pmatrix}$. What is the characteristic polynomial $p_A(\lambda) = \det(A - \lambda \mathbf{1} )$ in this case?
A1: $p_A(\lambda) = (2 - \lambda )(3 - \lambda ) - 1$
A2: $p_A(\lambda) = (2 - \lambda )(3 - \lambda ) + 1$
A3: $p_A(\lambda) = (2 - \lambda )(3 - \lambda ) $
A4: $p_A(\lambda) = (2 - \lambda )(3 - \lambda ) - (1-\lambda)(1-\lambda)$
A5: $p_A(\lambda) = (2 - \lambda )(1 - \lambda ) - 3$
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Last update: 2024-10