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Inverse Matrices 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 […]
  • Given All Eigenvalues and Eigenspaces, Compute a Matrix ProductGiven All Eigenvalues and Eigenspaces, Compute a Matrix Product Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces \[E_2=\Span\left \{\quad \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix} 1 \\ 2 \\ 1 \\ 1 […]
  • How to Obtain Information of a Vector if Information of Other Vectors are GivenHow to Obtain Information of a Vector if Information of Other Vectors are Given Let $A$ be a $3\times 3$ matrix and let \[\mathbf{v}=\begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}.\] Suppose that $A\mathbf{v}=-\mathbf{v}$ and $A\mathbf{w}=2\mathbf{w}$. Then find […]
  • 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 […]
  • If Two Matrices are Similar, then their Determinants are the SameIf Two Matrices are Similar, then their Determinants are the Same Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.   Proof. Suppose that $A$ and $B$ are similar. Then there exists a nonsingular matrix $S$ such that \[S^{-1}AS=B\] by definition. Then we […]
  • Top 10 Popular Math Problems in 2016-2017Top 10 Popular Math Problems in 2016-2017 It's been a year since I started this math blog!! More than 500 problems were posted during a year (July 19th 2016-July 19th 2017). I made a list of the 10 math problems on this blog that have the most views. Can you solve all of them? The level of difficulty among the top […]
  • Find a Linear Transformation Whose Image (Range) is a Given SubspaceFind a Linear Transformation Whose Image (Range) is a Given Subspace Let $V$ be the subspace of $\R^4$ defined by the equation \[x_1-x_2+2x_3+6x_4=0.\] Find a linear transformation $T$ from $\R^3$ to $\R^4$ such that the null space $\calN(T)=\{\mathbf{0}\}$ and the range $\calR(T)=V$. Describe $T$ by its matrix […]
  • Inverse Map of a Bijective Homomorphism is a Group HomomorphismInverse Map of a Bijective Homomorphism is a Group Homomorphism Let $G$ and $H$ be groups and let $\phi: G \to H$ be a group homomorphism. Suppose that $f:G\to H$ is bijective. Then there exists a map $\psi:H\to G$ such that \[\psi \circ \phi=\id_G \text{ and } \phi \circ \psi=\id_H.\] Then prove that $\psi:H \to G$ is also a group […]

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