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Square Roots of a Matrix Problems and Solutions


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  • Subspace of Skew-Symmetric Matrices and Its DimensionSubspace of Skew-Symmetric Matrices and Its Dimension Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.) (a) Prove that the subset $W$ is a subspace of $V$. (b) Find the […]
  • Rings $2\Z$ and $3\Z$ are Not IsomorphicRings $2\Z$ and $3\Z$ are Not Isomorphic Prove that the rings $2\Z$ and $3\Z$ are not isomorphic.   Definition of a ring homomorphism. Let $R$ and $S$ be rings. A homomorphism is a map $f:R\to S$ satisfying $f(a+b)=f(a)+f(b)$ for all $a, b \in R$, and $f(ab)=f(a)f(b)$ for all $a, b \in R$. A […]
  • Linear Properties of Matrix Multiplication and the Null Space of a MatrixLinear Properties of Matrix Multiplication and the Null Space of a Matrix Let $A$ be an $m \times n$ matrix. Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$. Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$. Then find $A\mathbf{w}$.   Hint. Recall that the null space of an […]
  • Similar Matrices Have the Same EigenvaluesSimilar Matrices Have the Same Eigenvalues Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same. Proof. We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]
  • A Group with a Prime Power Order Elements Has Order a Power of the Prime.A Group with a Prime Power Order Elements Has Order a Power of the Prime. Let $p$ be a prime number. Suppose that the order of each element of a finite group $G$ is a power of $p$. Then prove that $G$ is a $p$-group. Namely, the order of $G$ is a power of $p$. Hint. You may use Sylow's theorem. For a review of Sylow's theorem, please check out […]
  • Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is EvenEigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$. Then prove the following statements. (a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number. (b) The rank of $A$ is even.   Proof. (a) Each […]
  • Is a Set of All Nilpotent Matrix a Vector Space?Is a Set of All Nilpotent Matrix a Vector Space? Let $V$ denote the vector space of all real $n\times n$ matrices, where $n$ is a positive integer. Determine whether the set $U$ of all $n\times n$ nilpotent matrices is a subspace of the vector space $V$ or not.   Definition. An matrix $A$ is a nilpotent matrix if […]
  • If 2 by 2 Matrices Satisfy $A=AB-BA$, then $A^2$ is Zero MatrixIf 2 by 2 Matrices Satisfy $A=AB-BA$, then $A^2$ is Zero Matrix Let $A, B$ be complex $2\times 2$ matrices satisfying the relation \[A=AB-BA.\] Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.   Hint. Find the trace of $A$. Use the Cayley-Hamilton theorem Proof. We first calculate the […]

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