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Let T be a linear operator on a finite-dimensional vector space V. Refer to the definition of the determinant of \(\mathrm{T}\) on page 249 to prove the following results. (a) Prove that this definition is independent of the choice of an ordered basis for \(\mathrm{V}\). That is, prove that if \(\beta\) and \(\gamma\) are two ordered bases for \(\mathrm{V}\), then \(\operatorname{det}\left([\mathrm{T}]_{\beta}\right)=\operatorname{det}\left([\mathrm{T}]_{\gamma}\right)\). (b) Prove that \(T\) is invertible if and only if \(\operatorname{det}(T) \neq 0\). (c) Prove that if \(\mathrm{T}\) is invertible, then \(\operatorname{det}\left(\mathrm{T}^{-1}\right)=[\operatorname{det}(\mathrm{T})]^{-1}\). (d) Prove that if \(U\) is also a linear operator on \(V\), then \(\operatorname{det}(T U)=\) \(\operatorname{det}(T) \cdot \operatorname{det}(U)\).Sec. \(5.1\) Eigenvalues and Eigenvectors 259 (e) Prove that \(\operatorname{det}\left(T-\left.\lambda\right|_{V}\right)=\operatorname{det}\left([T]_{\beta}-\lambda I\right)\) for any scalar \(\lambda\) and any ordered basis \(\beta\) for \(\mathrm{V}\).

Let \(\mathrm{T}\) be a linear operator on a vector space with a \(\mathrm{T}\)-invariant subspace W. Prove that if \(v\) is an eigenvector of \(T_{W}\) with corresponding eigenvalue \(\lambda\), then \(v\) is also an eigenvector of \(\mathrm{T}\) with corresponding eigenvalue \(\lambda\).

(a) Prove that a linear operator \(\mathrm{T}\) on a finite-dimensional vector space is invertible if and only if zero is not an eigenvalue of \(T\). (b) Let \(\mathrm{T}\) be an invertible linear operator. Prove that a scalar \(\lambda\) is an eigenvalue of \(T\) if and only if \(\lambda^{-1}\) is an eigenvalue of \(T^{-1}\). (c) State and prove results analogous to (a) and (b) for matrices.

Each of the matrices that follow is a regular transition matrix for a three- state Markov chain. In all cases, the initial probability vector is $$ P=\left(\begin{array}{l} 0.3 \\ 0.3 \\ 0.4 \end{array}\right) $$ For each transition matrix, compute the proportions of objects in each state after two stages and the eventual proportions of objects in each state by determining the fixed probability vector. (a) \(\left(\begin{array}{rrr}0.6 & 0.1 & 0.1 \\ 0.1 & 0.9 & 0.2 \\ 0.3 & 0 & 0.7\end{array}\right)\) (b) \(\left(\begin{array}{lll}0.8 & 0.1 & 0.2 \\ 0.1 & 0.8 & 0.2 \\ 0.1 & 0.1 & 0.6\end{array}\right)\) (c) \(\left(\begin{array}{rrr}0.9 & 0.1 & 0.1 \\ 0.1 & 0.6 & 0.1 \\ 0 & 0.3 & 0.8\end{array}\right)\) (d) \(\left(\begin{array}{lll}0.4 & 0.2 & 0.2 \\ 0.1 & 0.7 & 0.2 \\ 0.5 & 0.1 & 0.6\end{array}\right)\) (e) \(\left(\begin{array}{lll}0.5 & 0.3 & 0.2 \\ 0.2 & 0.5 & 0.3 \\ 0.3 & 0.2 & 0.5\end{array}\right)\) (f) \(\left(\begin{array}{rrr}0.6 & 0 & 0.4 \\ 0.2 & 0.8 & 0.2 \\ 0.2 & 0.2 & 0.4\end{array}\right)\)

Let \(A\) be an \(n \times n\) matrix that is similar to an upper triangular matrix and has the distinct eigenvalues \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}\) with corresponding multiplicities \(m_{1}, m_{2}, \ldots, m_{k}\). Prove the following statements. (a) \(\operatorname{tr}(A)=\sum^{k} m_{i} \lambda_{i}\) (b) \(\operatorname{det}(A)=\left(\lambda_{1}\right)^{m_{1}}\left(\lambda_{2}\right)^{m_{2}} \ldots\left(\lambda_{k}\right)^{m_{k}}\).

(a) Prove that if \(A \in \mathrm{M}_{n \times n}(F)\) and the characteristic polynomial of \(A\) splits, then \(A\) is similar to an upper triangular matrix. (This proves the converse of Exercise \(9(\mathrm{~b}) .)\) Hint: Use mathematical induction on \(n\). For the general case, let \(v_{1}\) be an eigenvector of \(A\), and extend \(\left\\{v_{1}\right\\}\) to a basis \(\left\\{v_{1}, v_{2}, \ldots, v_{n}\right\\}\) for \(F^{n}\). Let \(P\) be the \(n \times n\) matrix whose \(j\) th column is \(v_{j}\), and consider \(P^{-1} A P\). Exercise 13(a) in Section \(5.1\) and Exercise 21 in Section \(4.3\) can be helpful. (b) Prove the converse of Exercise 9 (a). Visit goo.gl/gJSjRU for a solution.

A scalar matrix is a square matrix of the form \(\lambda I\) for some scalar \(\lambda\); that is, a scalar matrix is a diagonal matrix in which all the diagonal entries are equal. (a) Prove that if a square matrix \(A\) is similar to a scalar matrix \(\lambda I\), then \(A=\lambda I\). (b) Show that a diagonalizable matrix having only one eigenvalue is a scalar matrix. (c) Prove that \(\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right)\) is not diagonalizable.

(a) Prove that similar matrices have the same characteristic polynomial. (b) Show that the definition of the characteristic polynomial of a linear operator on a finite-dimensional vector space \(V\) is independent of the choice of basis for \(\mathrm{V}\).

Let \(A \in M_{n \times n}(F)\). Recall from Exercise 15 of Section \(5.1\) that \(A\) and \(A^{t}\) have the same characteristic polynomial and hence share the same eigenvalues with the same multiplicities. For any eigenvalue \(\lambda\) of \(A\) and \(A^{t}\), let \(\mathrm{E}_{\lambda}\) and \(\mathrm{E}_{\lambda}^{\prime}\) denote the corresponding eigenspaces for \(A\) and \(A^{t}\), respectively. (a) Show by way of example that for a given common eigenvalue, these two eigenspaces need not be the same. (b) Prove that for any eigenvalue \(\lambda, \operatorname{dim}\left(E_{\lambda}\right)=\operatorname{dim}\left(E_{\lambda}^{\prime}\right)\). (c) Prove that if \(A\) is diagonalizable, then \(A^{t}\) is also diagonalizable.

Use the Cayley-Hamilton theorem (Theorem 5.22) to prove its corollary for matrices. Warning: If \(f(t)=\operatorname{det}(A-t I)\) is the characteristic polynomial of \(A\), it is tempting to "prove" that \(f(A)=O\) by saying \(" f(A)=\operatorname{det}(A-A I)=\operatorname{det}(O)=0 . "\) Why is this argument incorrect? Visit goo.gl/ZMVn9i for a solution.