matrix-nonsingular-10seconds

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


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  • Problems and Solutions About Similar MatricesProblems and Solutions About Similar Matrices Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix. Prove the following statements. (a) If $A$ is similar to $B$, then $B$ is similar to $A$. (b) $A$ is similar to itself. (c) If $A$ is similar to $B$ and $B$ […]
  • Find Values of $a$ so that the Matrix is NonsingularFind Values of $a$ so that the Matrix is Nonsingular Let $A$ be the following $3 \times 3$ matrix. \[A=\begin{bmatrix} 1 & 1 & -1 \\ 0 &1 &2 \\ 1 & 1 & a \end{bmatrix}.\] Determine the values of $a$ so that the matrix $A$ is nonsingular.   Solution. We use the fact that a matrix is nonsingular if and only if […]
  • The Set of Vectors Perpendicular to a Given Vector is a SubspaceThe Set of Vectors Perpendicular to a Given Vector is a Subspace Fix the row vector $\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$, and let $\R^3$ be the vector space of $3 \times 1$ column vectors. Define \[W = \{ \mathbf{v} \in \R^3 \mid \mathbf{b} \mathbf{v} = 0 \}.\] Prove that $W$ is a vector subspace of $\R^3$.   […]
  • The Preimage of Prime ideals are Prime IdealsThe Preimage of Prime ideals are Prime Ideals Let $f: R\to R'$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R'$. Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.   Proof. The preimage of an ideal by a ring homomorphism is an ideal. (See the post "The inverse image of an ideal by […]
  • Solving a System of Linear Equations By Using an Inverse MatrixSolving a System of Linear Equations By Using an Inverse Matrix Consider the system of linear equations \begin{align*} x_1&= 2, \\ -2x_1 + x_2 &= 3, \\ 5x_1-4x_2 +x_3 &= 2 \end{align*} (a) Find the coefficient matrix and its inverse matrix. (b) Using the inverse matrix, solve the system of linear equations. (The Ohio […]
  • Find Values of $a$ so that Augmented Matrix Represents a Consistent SystemFind Values of $a$ so that Augmented Matrix Represents a Consistent System Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations. \[A= \left[\begin{array}{rrr|r} 1 & 2 & 3 & 4 \\ 2 &-1 & -2 & a^2 \\ -1 & -7 & -11 & a \end{array} \right],\] where $a$ is a real number. Determine all the […]
  • Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly IndependentUsing the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent By calculating the Wronskian, determine whether the set of exponential functions \[\{e^x, e^{2x}, e^{3x}\}\] is linearly independent on the interval $[-1, 1]$.   Solution. The Wronskian for the set $\{e^x, e^{2x}, e^{3x}\}$ is given […]
  • The Cyclotomic Field of 8-th Roots of Unity is $\Q(\zeta_8)=\Q(i, \sqrt{2})$The Cyclotomic Field of 8-th Roots of Unity is $\Q(\zeta_8)=\Q(i, \sqrt{2})$ Let $\zeta_8$ be a primitive $8$-th root of unity. Prove that the cyclotomic field $\Q(\zeta_8)$ of the $8$-th root of unity is the field $\Q(i, \sqrt{2})$.   Proof. Recall that the extension degree of the cyclotomic field of $n$-th roots of unity is given by […]

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