# nilpotent-matrix

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• Find 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 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 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 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 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 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 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 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 […]