Matrices & Determinants (Advanced)

Matrices & Determinants: core idea

• 1. Cayley–Hamilton: for $A=\begin{pmatrix}1&1\\0&1\end{pmatrix}$, $A^2-2A+I$ equals zero matrix? $1/0$

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• 2. If $A$ diagonalizable $D=\mathrm{diag}(2,3)$, $A^{2}$ eigenvalues product?

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• 3. Rank of $\begin{pmatrix}1&2&3\\2&4&6\\0&0&1\end{pmatrix}$?

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• 4. Det of $\begin{pmatrix}1&2&0\\0&3&0\\0&0&4\end{pmatrix}$?

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• 5. Nilpotent $N\ne0$, $N^2=0$ for $2\times2$: possible det?

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• 6. Minimal polynomial degree of $\begin{pmatrix}2&0\\0&2\end{pmatrix}$?

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• 7. Exp of zero matrix: det?

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• 8. System $Ax=0$ nontrivial iff det $A$?

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• 9. If eigenvalues $1,2,3$, det $A$?

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