Let $A \in \mathcal{M}_{2 \times 2} (\mathbb{R})$ such that $A^2+\mathbb{I}_{2 \times 2}=0$. Evaluate the minimal polynomial of $A$ and prove that $A$ is not diagonizable.
Solution
The matrix $A$ satisfies the equation $x^2+1$. The polynomial $x^2+1$ is irreducible over $\mathbb{R}[x]$. Thus the minimal polynomial of $A$ is $\mathfrak{m}_A(x)=x^2+1$. If $A$ was diagonizable that would mean that the minimal polynomial of $A$ would have exactly two roots over $\mathbb{R}$ which is an obscurity. Hence $A$ is not diagonizable.
Solution
The matrix $A$ satisfies the equation $x^2+1$. The polynomial $x^2+1$ is irreducible over $\mathbb{R}[x]$. Thus the minimal polynomial of $A$ is $\mathfrak{m}_A(x)=x^2+1$. If $A$ was diagonizable that would mean that the minimal polynomial of $A$ would have exactly two roots over $\mathbb{R}$ which is an obscurity. Hence $A$ is not diagonizable.
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