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Nilpotent Matrix Problems and Solutions


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  • Find Values of $a$ so that the Matrix is NonsingularFind Values of $a$ so that the Matrix is Nonsingular Let $A$ be the following $3 \times 3$ matrix. \[A=\begin{bmatrix} 1 & 1 & -1 \\ 0 &1 &2 \\ 1 & 1 & a \end{bmatrix}.\] Determine the values of $a$ so that the matrix $A$ is nonsingular.   Solution. We use the fact that a matrix is nonsingular if and only if […]
  • If $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent MatrixIf $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent Matrix Let $A$ be a square matrix such that \[A^{\trans}A=A,\] where $A^{\trans}$ is the transpose matrix of $A$. Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix.     Hint. Recall the basic properties of transpose […]
  • Linear Transformation to 1-Dimensional Vector Space and Its KernelLinear Transformation to 1-Dimensional Vector Space and Its Kernel Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation. Prove the followings. (a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$. (b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the […]
  • Calculate Determinants of MatricesCalculate Determinants of Matrices Calculate the determinants of the following $n\times n$ matrices. \[A=\begin{bmatrix} 1 & 0 & 0 & \dots & 0 & 0 &1 \\ 1 & 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 1 & 1 & \dots & 0 & 0 & 0 \\ \vdots & \vdots […]
  • Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4 Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.   Hint. Use Sylow's theorem. (See Sylow’s Theorem (Summary) for a review of Sylow's theorem.) Recall that if there is a unique Sylow $p$-subgroup in a group $GH$, then it is […]
  • The Matrix for the Linear Transformation of the Reflection Across a Line in the PlaneThe Matrix for the Linear Transformation of the Reflection Across a Line in the Plane Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$. Then find the matrix representation of the linear transformation $T$ with respect to the […]
  • Order of Product of Two Elements in a GroupOrder of Product of Two Elements in a Group Let $G$ be a group. Let $a$ and $b$ be elements of $G$. If the order of $a, b$ are $m, n$ respectively, then is it true that the order of the product $ab$ divides $mn$? If so give a proof. If not, give a counterexample.   Proof. We claim that it is not true. As a […]
  • Quiz 12. Find Eigenvalues and their Algebraic and Geometric MultiplicitiesQuiz 12. Find Eigenvalues and their Algebraic and Geometric Multiplicities (a) Let \[A=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 &1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix}.\] Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue. (b) Let \[A=\begin{bmatrix} 0 & 0 & 0 & 0 […]

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