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- Find All Values of $a$ which Will Guarantee that $A$ Has Eigenvalues 0, 3, and -3. Let $A$ be the matrix given by \[ A= \begin{bmatrix} -2 & 0 & 1 \\ -5 & 3 & a \\ 4 & -2 & -1 \end{bmatrix} \] for some variable $a$. Find all values of $a$ which will guarantee that $A$ has eigenvalues $0$, $3$, and $-3$. Solution. Let $p(t)$ be the […]
- The set of $2\times 2$ Symmetric Matrices is a Subspace Let $V$ be the vector space over $\R$ of all real $2\times 2$ matrices. Let $W$ be the subset of $V$ consisting of all symmetric matrices. (a) Prove that $W$ is a subspace of $V$. (b) Find a basis of $W$. (c) Determine the dimension of $W$. Proof. […]
- Given the Data of Eigenvalues, Determine if the Matrix is Invertible In each of the following cases, can we conclude that $A$ is invertible? If so, find an expression for $A^{-1}$ as a linear combination of positive powers of $A$. If $A$ is not invertible, explain why not. (a) The matrix $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda=i , […]
- Compute $A^5\mathbf{u}$ Using Linear Combination Let \[A=\begin{bmatrix} -4 & -6 & -12 \\ -2 &-1 &-4 \\ 2 & 3 & 6 \end{bmatrix}, \quad \mathbf{u}=\begin{bmatrix} 6 \\ 5 \\ -3 \end{bmatrix}, \quad \mathbf{v}=\begin{bmatrix} -2 \\ 0 \\ 1 \end{bmatrix}, \quad \text{ and } […]
- If Two Vectors Satisfy $A\mathbf{x}=0$ then Find Another Solution Suppose that the vectors \[\mathbf{v}_1=\begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} -4 \\ 0 \\ -3 \\ -2 \\ 1 \end{bmatrix}\] are a basis vectors for the null space of a $4\times 5$ […]
- Linear Transformation $T:\R^2 \to \R^2$ Given in Figure Let $T:\R^2\to \R^2$ be a linear transformation such that it maps the vectors $\mathbf{v}_1, \mathbf{v}_2$ as indicated in the figure below. Find the matrix representation $A$ of the linear transformation $T$. Solution 1. From the figure, we see […]
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