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## Chapter 10 Exercise B

1. Solution. Because $T_\mathbb{C}$ has no real eigenvalues, if $\lambda$ is an eigenvalue of $T_\mathbb{C}$, then $\overline{\lambda}$ is also an eigenvalue of $T_\mathbb{C}$ with equal multiplicity. The determinant of $T_\mathbb{C}$,…

## Chapter 10 Exercise A

1. Solution: If $T$ is invertible, then there exists $S\in\ca L(V)$ such that $TS=ST=I$. Then it follows from 10.4 that$\ca M(S,(v_1,\cdots,v_n))\ca M(T,(v_1,\cdots,v_n))=\ca M(ST,(v_1,\cdots,v_n))=I$$\ca M(T,(v_1,\cdots,v_n))\ca M(S,(v_1,\cdots,v_n))=\ca M(TS,(v_1,\cdots,v_n))=I.$Hence $\ca M(T,(v_1,\cdots,v_n))$ is invertible.If…