Find Values of $a, b, c$ such that the Given Matrix is Diagonalizable

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• Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix Let $A$ be an $n\times n$ matrix with the characteristic polynomial \[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\] Assume that the matrix $A$ is diagonalizable. (a) Find the size of the matrix $A$. (b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue […]
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• How to Calculate and Simplify a Matrix Polynomial Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$. Calculate and simplify the expression \[-T^3+4T^2+5T-2I,\] where $I$ is the $3\times 3$ identity matrix. (The Ohio State University Linear Algebra Exam) Hint. Use the […]
• Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$ Let \[A=\begin{bmatrix} 1 & -14 & 4 \\ -1 &6 &-2 \\ -2 & 24 & -7 \end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix} 4 \\ -1 \\ -7 \end{bmatrix}.\] Find $A^{10}\mathbf{v}$. You may use the following information without proving […]
• Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue Let \[A=\begin{bmatrix} 1 & 2 & 1 \\ -1 &4 &1 \\ 2 & -4 & 0 \end{bmatrix}.\] The matrix $A$ has an eigenvalue $2$. Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$. (The Ohio State University, Linear Algebra Final Exam […]
• Diagonalize a 2 by 2 Matrix if Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}\] is diagonalizable. If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
• Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$ Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities. What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix? (The Ohio State University, Linear Algebra Final Exam […]
• Given All Eigenvalues and Eigenspaces, Compute a Matrix Product Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces \[E_2=\Span\left \{\quad \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix} 1 \\ 2 \\ 1 \\ 1 […]