The Determinant of a Skew-Symmetric Matrix is Zero

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• Subspaces of Symmetric, Skew-Symmetric Matrices Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$. (a) The set $S$ consisting of all $n\times n$ symmetric matrices. (b) The set $T$ consisting of […]
• If $A$ is a Skew-Symmetric Matrix, then $I+A$ is Nonsingular and $(I-A)(I+A)^{-1}$ is Orthogonal Let $A$ be an $n\times n$ real skew-symmetric matrix. (a) Prove that the matrices $I-A$ and $I+A$ are nonsingular. (b) Prove that \[B=(I-A)(I+A)^{-1}\] is an orthogonal matrix.   Proof. (a) Prove that the matrices $I-A$ and $I+A$ are nonsingular. The […]
• Eigenvalues 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 […]
• Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation Let $V$ be the vector space of all $n\times n$ real matrices. Let us fix a matrix $A\in V$. Define a map $T: V\to V$ by \[ T(X)=AX-XA\] for each $X\in V$. (a) Prove that $T:V\to V$ is a linear transformation. (b) Let $B$ be a basis of $V$. Let $P$ be the matrix […]
• The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$ Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix. Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula: \[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\] Using the formula, calculate […]
• Eigenvalues and Eigenvectors of The Cross Product Linear Transformation We fix a nonzero vector $\mathbf{a}$ in $\R^3$ and define a map $T:\R^3\to \R^3$ by \[T(\mathbf{v})=\mathbf{a}\times \mathbf{v}\] for all $\mathbf{v}\in \R^3$. Here the right-hand side is the cross product of $\mathbf{a}$ and $\mathbf{v}$. (a) Prove that $T:\R^3\to \R^3$ is […]
• Express a Hermitian Matrix as a Sum of Real Symmetric Matrix and a Real Skew-Symmetric Matrix Recall that a complex matrix is called Hermitian if $A^*=A$, where $A^*=\bar{A}^{\trans}$. Prove that every Hermitian matrix $A$ can be written as the sum \[A=B+iC,\] where $B$ is a real symmetric matrix and $C$ is a real skew-symmetric matrix.   Proof. Since […]
• Quiz 11. Find Eigenvalues and Eigenvectors/ Properties of Determinants (a) Find all the eigenvalues and eigenvectors of the matrix \[A=\begin{bmatrix} 3 & -2\\ 6& -4 \end{bmatrix}.\] (b) Let \[A=\begin{bmatrix} 1 & 0 & 3 \\ 4 &5 &6 \\ 7 & 0 & 9 \end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 &0 […]