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Math-Magic


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  • True or False Problems on Midterm Exam 1 at OSU Spring 2018True or False Problems on Midterm Exam 1 at OSU Spring 2018 The following problems are True or False. Let $A$ and $B$ be $n\times n$ matrices. (a) If $AB=B$, then $B$ is the identity matrix. (b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions. (c) If $A$ […]
  • Commuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same EigenvaluesCommuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same Eigenvalues Let $A$ and $B$ be square matrices such that they commute each other: $AB=BA$. Assume that $A-B$ is a nilpotent matrix. Then prove that the eigenvalues of $A$ and $B$ are the same.   Proof. Let $N:=A-B$. By assumption, the matrix $N$ is nilpotent. This […]
  • 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 […]
  • Conditions on Coefficients that a Matrix is NonsingularConditions on Coefficients that a Matrix is Nonsingular (a) Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that the entries of the matrix $A$ satisfy the following relation. \[|a_{ii}|>|a_{i1}|+\cdots +|a_{i\,i-1}|+|a_{i \, i+1}|+\cdots +|a_{in}|\] for all $1 \leq i \leq n$. Show that the matrix $A$ is nonsingular. (b) Let […]
  • If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$ Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.   Proof. As non-singularity and invertibility are equivalent, we know that $M$ has the inverse matrix $M^{-1}$. Let us think backwards. Suppose that […]
  • The Ideal Generated by a Non-Unit Irreducible Element in a PID is MaximalThe Ideal Generated by a Non-Unit Irreducible Element in a PID is Maximal Let $R$ be a principal ideal domain (PID). Let $a\in R$ be a non-unit irreducible element. Then show that the ideal $(a)$ generated by the element $a$ is a maximal ideal.   Proof. Suppose that we have an ideal $I$ of $R$ such that \[(a) \subset I \subset […]
  • Diagonalize a 2 by 2 Symmetric MatrixDiagonalize a 2 by 2 Symmetric Matrix Diagonalize the $2\times 2$ matrix $A=\begin{bmatrix} 2 & -1\\ -1& 2 \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   Solution. The characteristic polynomial $p(t)$ of the matrix $A$ […]
  • If Two Vectors Satisfy $A\mathbf{x}=0$ then Find Another SolutionIf 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 $4\times 5$ […]

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