8b03f4…: “Over the cloudy sphere”

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Determine the characteristic polynomial of $A = \begin {bmatrix} -2 &3 \\1&1\end {bmatrix}$ .
$\answer {1}\lambda ^2 +\answer {1}\lambda +\answer {-5}$

The characteristic polynomial of a square matrix $A$ is $\det (A-\lambda I)$ .

Let $A= \begin {bmatrix} 2 &5 \\5&2\end {bmatrix}$ .

(a): Determine the characteristic polynomial of $A$ .

$\answer {1}\lambda ^2 +\answer {-4}\lambda +\answer {-21}$
(b): What are the real eigenvalues of $A$ ?

List from smallest to largest: $\lambda _1 = \answer {-3}$ , $\lambda _2 = \answer {7}$

Let $A= \begin {bmatrix} 1&5&4 \\0&3&-4\\ 0&0&-2\end {bmatrix}$ .

(a): Determine the characteristic polynomial of $A$ .

-6 $(\lambda -1)(\lambda -3)(\lambda +2)$ $(1-\lambda )(3-\lambda )(-2-\lambda )$ $(1-\lambda )(3-\lambda )(2-\lambda )$
(b): What are the real eigenvalues of $A$ ?

List from smallest to largest: $\lambda _1 = \answer {-2}$ , $\lambda _2 = \answer {1}$ , $\lambda _3 = \answer {3}$

If $A$ is triangular, then so is $A-\lambda I$ and there is an easy way to determine the determinant of triangular matrices.

Let $A= \begin {bmatrix} 1&2&0 \\3&-3&0\\ 0&1&-2\end {bmatrix}$ .

(a): Determine the characteristic polynomial of $A$ .

$(-2-\lambda )(\answer {1}\lambda ^2+\answer {2}\lambda +\answer {-9})$
(b): What are the real eigenvalues of $A$ ?

$\lambda = \answer {-2}, \lambda = \answer {-1}\pm \sqrt {\answer {10}}$

Try cofactor expansion. Also, quadratic formula.

In the characteristic polynomial, $(3-\lambda )(4-\lambda )^3(-3-\lambda )(3-\lambda )$ what is the multiplicity of $\dots$

(a): $\lambda = 3$ ? $\answer {2}$
(b): $\lambda = 4$ ? $\answer {3}$
(c): $\lambda = -3$ ? $\answer {1}$

True/False: Let $A$ be an $n\times n$ matrix. $A$ is invertible if and only if zero is an eigenvalue of $A$ .

True False