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Let \(A \in M(n, \mathbb{F})\) have an eigenvalue \(\lambda\) with corresponding eigenvector \(v\). True or False
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Let \(A \in M(n, \mathbb{F})\) have an eigenvalue \(\lambda\) with corresponding eigenvector \(v\). True or FalseLet \(A \in M(n, \mathbb{F})\) have an eigenvalue \(\lambda\) with corresponding eigenvector \(v\). True or False a) \(-v\) is an eigenvector of \(-A\ ...
by MathsGee Platinum
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asked in Mathematics 11 minutes ago
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Let \(C\) be a \(2 \times 2\) matrix of real numbers. Give a proof or counterexample to each of the following assertions:
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Let \(C\) be a \(2 \times 2\) matrix of real numbers. Give a proof or counterexample to each of the following assertions:Let \(C\) be a \(2 \times 2\) matrix of real numbers. Give a proof or counterexample to each of the following assertions: a) \(\operatorname{det}\left ...
by MathsGee Platinum
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asked in Mathematics 12 minutes ago
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A \(3 \times 3\) real matrix need not have any real eigenvalues.
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A \(3 \times 3\) real matrix need not have any real eigenvalues.True or False - and Why? a) A \(3 \times 3\) real matrix need not have any real eigenvalues. b) If an \(n \times n\) matrix \(A\) is invertible, then ...
by MathsGee Platinum
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asked in Mathematics 13 minutes ago
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Let \(A\) be a square matrix. Proof or Counterexample. If \(A\) is diagonalizable, then so is \(A^{2}\).
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Let \(A\) be a square matrix. Proof or Counterexample. If \(A\) is diagonalizable, then so is \(A^{2}\).Let \(A\) be a square matrix. Proof or Counterexample. a) If \(A\) is diagonalizable, then so is \(A^{2}\). b) If \(A^{2}\) is diagonalizable, then so ...
by MathsGee Platinum
(104,008 points)
asked in Mathematics 15 minutes ago
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Find an orthogonal matrix \(R\) that diagonalizes \(A:=\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2\end{array}\right)\)
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Let \(A=\left(\begin{array}{cc}a & b-a \\ 0 & b\end{array}\right)\) - Diagonalize \(A\).
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Let \(A=\left(\begin{array}{cc}a & b-a \\ 0 & b\end{array}\right)\) - Diagonalize \(A\).Let \(A=\left(\begin{array}{cc}a & b-a \\ 0 & b\end{array}\right)\) a) Diagonalize \(A\). b) Use this to compute \(A^{k}\) for any integer \( ...
by MathsGee Platinum
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asked in Mathematics 20 minutes ago
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If a matrix \(A\) is diagonalizable, show that for any scalar \(c\) so is the matrix \(A+c I\).
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Diagonalize the matrix \[ A=\left(\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right) \]
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Diagonalize the matrix \[ A=\left(\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right) \]Diagonalize the matrix \ A=\left(\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right) \ by finding the ...
by MathsGee Platinum
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asked in Mathematics 22 minutes ago
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Let \(A=\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2\end{array}\right)\).
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Let \(A=\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2\end{array}\right)\).Let \(A=\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2\end{array}\right)\). a) What is the dimension of the image ...
by MathsGee Platinum
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asked in Mathematics 22 minutes ago
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If \(A\) and \(B\) can be simultaneously diagonalized, show that \(A B=B A\).
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If \(A\) and \(B\) can be simultaneously diagonalized, show that \(A B=B A\).Two matrices \(A, B\) can be simultaneously diagonalized if there is an invertible matrix that diagonalizes both of them. In other words, if there is ...
by MathsGee Platinum
(104,008 points)
asked in Mathematics 24 minutes ago
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What can you say about the eigenvalues and eigenvectors of \(A^{-1} ?\) Justify your response.
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What can you say about the eigenvalues and eigenvectors of \(A^{-1} ?\) Justify your response.Let \(A\) be an invertible matrix with eigenvalues \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}\) and corresponding eigenvectors \(\vec{v}_{1}, \ve ...
by MathsGee Platinum
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asked in Mathematics 26 minutes ago
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Let \(A\) be a square matrix. If the eigenvectors \(v_{1}, \ldots v_{k}\) have distinct eigenvalues, show that these vectors are linearly independent.

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Give an example of a matrix \(A\) with the following three properties:
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Give an example of a matrix \(A\) with the following three properties:Give an example of a matrix \(A\) with the following three properties: i). \(A\) has eigenvalues \(-1\) and 2 . ii). The eigenvalue \(-1\) has eigenve ...
by MathsGee Platinum
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asked in Mathematics 27 minutes ago
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Compute the determinant of \(A^{10}+A\).
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Compute the determinant of \(A^{10}+A\).a) Find a \(2 \times 2\) real matrix \(A\) that has an eigenvalue \(\lambda_{1}=1\) with eigenvector \(E_{1}=\) \(\left(\begin{array}{l}1 \\ 2\end{arr ...
by MathsGee Platinum
(104,008 points)
asked in Mathematics 28 minutes ago
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[Frobenius] Let \(A, B\), and \(C\) be matrices so that the products \(A B\) and \(B C\) are defined.
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[Frobenius] Let \(A, B\), and \(C\) be matrices so that the products \(A B\) and \(B C\) are defined.Frobenius Let \(A, B\), and \(C\) be matrices so that the products \(A B\) and \(B C\) are defined. Use the obvious \ \operatorname{dim}\left(\left ...
by MathsGee Platinum
(104,008 points)
asked in Mathematics 29 minutes ago
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Let \(U \subset V\) and \(W\) be finite dimensional linear spaces and \(L: V \rightarrow W\) a linear map. Show that
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Let \(U \subset V\) and \(W\) be finite dimensional linear spaces and \(L: V \rightarrow W\) a linear map. Show thatLet \(U \subset V\) and \(W\) be finite dimensional linear spaces and \(L: V \rightarrow W\) a linear map. Show that \ \operatorname{dim}\left(\left. ...
by MathsGee Platinum
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asked in Mathematics 31 minutes ago
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Let \(A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}\) and \(B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}\).
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Let \(A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}\) and \(B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}\).Let \(A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}\) and \(B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}\). Prove that \(\operatorname{rank} A+\o ...
by MathsGee Platinum
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asked in Mathematics 31 minutes ago
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Let \(A(t)\) be a family of real square matrices depending smoothly on the parameter \(t\).
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Let \(A(t)\) be a family of real square matrices depending smoothly on the parameter \(t\).Let \(A(t)\) be a family of real square matrices depending smoothly on the parameter \(t\). a) Find a formula for the derivative of \(A^{2}(t)\). b) ...
by MathsGee Platinum
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asked in Mathematics 32 minutes ago
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Let \(A(t)\) be a family of invertible real matrices depending on the real parameter \(t\) and assume they are invertible.
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Let \(A(t)\) be a family of invertible real matrices depending on the real parameter \(t\) and assume they are invertible.Let \(A(t)\) be a family of invertible real matrices depending on the real parameter \(t\) and assume they are invertible. Show that the inverse matri ...
by MathsGee Platinum
(104,008 points)
asked in Mathematics 33 minutes ago
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