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Inverse Matrices Problems and Solutions


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  • An Example of a Real Matrix that Does Not Have Real EigenvaluesAn Example of a Real Matrix that Does Not Have Real Eigenvalues Let \[A=\begin{bmatrix} a & b\\ -b& a \end{bmatrix}\] be a $2\times 2$ matrix, where $a, b$ are real numbers. Suppose that $b\neq 0$. Prove that the matrix $A$ does not have real eigenvalues.   Proof. Let $\lambda$ be an arbitrary eigenvalue of […]
  • $(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.$(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain. Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.   Proof. Consider the ring $R[t]$, where $t$ is a variable. Since $R$ is an integral domain, so is $R[t]$. Define the function $\Psi:R[x,y] \to R[t]$ sending […]
  • Unit Vectors and Idempotent MatricesUnit Vectors and Idempotent Matrices A square matrix $A$ is called idempotent if $A^2=A$. (a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$. Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$. Prove that $P$ is an idempotent matrix. (b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be […]
  • Determine Null Spaces of Two MatricesDetermine Null Spaces of Two Matrices Let \[A=\begin{bmatrix} 1 & 2 & 2 \\ 2 &3 &2 \\ -1 & -3 & -4 \end{bmatrix} \text{ and } B=\begin{bmatrix} 1 & 2 & 2 \\ 2 &3 &2 \\ 5 & 3 & 3 \end{bmatrix}.\] Determine the null spaces of matrices $A$ and $B$.   Proof. The null space of the […]
  • Two Eigenvectors Corresponding to Distinct Eigenvalues are Linearly IndependentTwo Eigenvectors Corresponding to Distinct Eigenvalues are Linearly Independent Let $A$ be an $n\times n$ matrix. Suppose that $\lambda_1, \lambda_2$ are distinct eigenvalues of the matrix $A$ and let $\mathbf{v}_1, \mathbf{v}_2$ be eigenvectors corresponding to $\lambda_1, \lambda_2$, respectively. Show that the vectors $\mathbf{v}_1, \mathbf{v}_2$ are […]
  • Find All the Eigenvalues of 4 by 4 MatrixFind All the Eigenvalues of 4 by 4 Matrix Find all the eigenvalues of the matrix \[A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 &0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}.\] (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. We compute the […]
  • Find a Basis for the Subspace spanned by Five VectorsFind a Basis for the Subspace spanned by Five Vectors Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where \[ \mathbf{v}_{1}= \begin{bmatrix} 1 \\ 2 \\ 2 \\ -1 \end{bmatrix} ,\;\mathbf{v}_{2}= \begin{bmatrix} 1 \\ 3 \\ 1 \\ 1 \end{bmatrix} ,\;\mathbf{v}_{3}= \begin{bmatrix} 1 \\ 5 \\ -1 […]
  • Determine When the Given Matrix InvertibleDetermine When the Given Matrix Invertible For which choice(s) of the constant $k$ is the following matrix invertible? \[A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &2 &k \\ 1 & 4 & k^2 \end{bmatrix}.\]   (Johns Hopkins University, Linear Algebra Exam)   Hint. An $n\times n$ matrix is […]

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