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• Find the Formula for the Power of a Matrix Let $A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &0 &1 \\ 0 & 0 & 1 \end{bmatrix}$ be a $3\times 3$ matrix. Then find the formula for $A^n$ for any positive integer $n$.   Proof. We first compute several powers of $A$ and guess the general formula. We […]
• The Zero is the only Nilpotent Element of the Quotient Ring by its Nilradical Prove that if $R$ is a commutative ring and $\frakN(R)$ is its nilradical, then the zero is the only nilpotent element of $R/\frakN(R)$. That is, show that $\frakN(R/\frakN(R))=0$. Proof. Let $r\in R$ and if $x:=r+\frakN(R) \in R/\frakN(R)$ is a nilpotent element of […]
• Show that the Given 2 by 2 Matrix is Singular Consider the matrix $M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$. (a) Show that $M$ is singular. (b) Find a non-zero vector $\mathbf{v}$ such that $M \mathbf{v} = \mathbf{0}$, where $\mathbf{0}$ is the $2$-dimensional zero vector.   Solution. (a) Show […]
• If a Symmetric Matrix is in Reduced Row Echelon Form, then Is it Diagonal? Recall that a matrix $A$ is symmetric if $A^\trans = A$, where $A^\trans$ is the transpose of $A$. Is it true that if $A$ is a symmetric matrix and in reduced row echelon form, then $A$ is diagonal? If so, prove it. Otherwise, provide a counterexample.   Proof. […]
• Common Eigenvector of Two Matrices and Determinant of Commutator Let $A$ and $B$ be $n\times n$ matrices. Suppose that these matrices have a common eigenvector $\mathbf{x}$. Show that $\det(AB-BA)=0$. Steps. Write down eigenequations of $A$ and $B$ with the eigenvector $\mathbf{x}$. Show that AB-BA is singular. A matrix is […]
• Using Properties of Inverse Matrices, Simplify the Expression Let $A, B, C$ be $n\times n$ invertible matrices. When you simplify the expression $C^{-1}(AB^{-1})^{-1}(CA^{-1})^{-1}C^2,$ which matrix do you get? (a) $A$ (b) $C^{-1}A^{-1}BC^{-1}AC^2$ (c) $B$ (d) $C^2$ (e) $C^{-1}BC$ (f) $C$   Solution. In this problem, we […]
• Invertible Idempotent Matrix is the Identity Matrix A square matrix $A$ is called idempotent if $A^2=A$. Show that a square invertible idempotent matrix is the identity matrix. Proof. Let $A$ be an $n \times n$ invertible idempotent matrix. Since $A$ is invertible, the inverse matrix $A^{-1}$ of $A$ exists and it […]
• Algebraic Number is an Eigenvalue of Matrix with Rational Entries A complex number $z$ is called algebraic number (respectively, algebraic integer) if $z$ is a root of a monic polynomial with rational (respectively, integer) coefficients. Prove that $z \in \C$ is an algebraic number (resp. algebraic integer) if and only if $z$ is an eigenvalue of […]

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