matrix-nonsingular-10seconds

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How to Prove a Matrix is Nonsingular in 10 Seconds!!


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  • The Sum of Subspaces is a Subspace of a Vector SpaceThe Sum of Subspaces is a Subspace of a Vector Space Let $V$ be a vector space over a field $K$. If $W_1$ and $W_2$ are subspaces of $V$, then prove that the subset \[W_1+W_2:=\{\mathbf{x}+\mathbf{y} \mid \mathbf{x}\in W_1, \mathbf{y}\in W_2\}\] is a subspace of the vector space $V$.   Proof. We prove the […]
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  • Is the Product of a Nilpotent Matrix and an Invertible Matrix Nilpotent?Is the Product of a Nilpotent Matrix and an Invertible Matrix Nilpotent? A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix. (a) If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent. (b) Let $P$ […]
  • Determine Trigonometric Functions with Given ConditionsDetermine Trigonometric Functions with Given Conditions (a) Find a function \[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)\] such that $g(0) = g(\pi/2) = g(\pi) = 0$, where $a, b, c$ are constants. (b) Find real numbers $a, b, c$ such that the function \[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 […]
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  • How Many Solutions for $x+x=1$ in a Ring?How Many Solutions for $x+x=1$ in a Ring? Is there a (not necessarily commutative) ring $R$ with $1$ such that the equation \[x+x=1 \] has more than one solutions $x\in R$?   Solution. We claim that there is at most one solution $x$ in the ring $R$. Suppose that we have two solutions $r, s \in R$. That is, we […]
  • Transpose of a Matrix and Eigenvalues and Related QuestionsTranspose of a Matrix and Eigenvalues and Related Questions Let $A$ be an $n \times n$ real matrix. Prove the followings. (a) The matrix $AA^{\trans}$ is a symmetric matrix. (b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal. (c) The matrix $AA^{\trans}$ is non-negative definite. (An $n\times n$ […]
  • Prove that $\{ 1 , 1 + x , (1 + x)^2 \}$ is a Basis for the Vector Space of Polynomials of Degree $2$ or LessProve that $\{ 1 , 1 + x , (1 + x)^2 \}$ is a Basis for the Vector Space of Polynomials of Degree $2$ or Less Let $\mathbf{P}_2$ be the vector space of polynomials of degree $2$ or less. (a) Prove that the set $\{ 1 , 1 + x , (1 + x)^2 \}$ is a basis for $\mathbf{P}_2$. (b) Write the polynomial $f(x) = 2 + 3x - x^2$ as a linear combination of the basis $\{ 1 , 1+x , (1+x)^2 […]

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