10 questions linked to/from Rank of skew-symmetric matrix
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### Every skew-symmetric matrix has even rank [duplicate]

Let $F$ be a field where $char(F)\neq2$ and let $A$ be a skew-symmetric matrix over $F$. Prove that rank of $A$ is even. I think the best way to prove it, is using induction on size of $A$. for $n=1$...
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### Prove that the rank of a real skew symmetric matrix is not $1$

I was thinking of using the RREF of $A$, where $A$ is an $n \times n$ skew symmetric matrix. So, suppose that it's rank is 1. then the first row has at least one non zero element(which is 1), while ...
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### Show that rank of skew-symetric is even number

$$A = -A^T$$ I assume that $A$ is not singular. So $$\det{A} \neq 0$$ Then $$\det(A) = \det(-A^T) = \det(-I_{n} A^T) = (-1)^n\det(A^T) = (-1)^n\det(A)$$ So I get that $n$ must be even. But what ...
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### Proof Verification: Skew-symmetric matrix has even rank

I thought of this argument to show that the rank of a skew-symmetric matrix is even. I'd appreciate it if someone could look over it and let me know whether or not it looks correct. Let $A$ be an $n$...
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### Finding rank of matrix

Suppose $B$ is a non-zero real skew-symmetric matrix of order $3$ and $A$ is a non-singular matrix with inverse $C$. Then rank of $ABC$ is: (A) $0, 1, 2$ (B) definitely $1$ (C) definitely $2$ (D) ...
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### Let $F$ be a field such that 1+1 is not equal to zero. Let $A$ element of $M_n(F)$ such that $A^T = -A$.

Let $F$ be a field such that 1+1 is not equal to zero. Let $A$ element of $M_n(F)$ such that $A^T = -A$. a) Show that if $A$ is non-singular, then $n$ is even. b) Show that if $A$ is singular, ...
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### $vv^TA-Avv^T$ where $A$ is skew-symmetric and $vv^T$ is rank one

Suppose I have the following: $$vv^TA-Avv^T$$ $v\in \mathbb{R}^{n}$ with $\|v\|_2=1$, so $vv^T$ is PSD, rank one and $\operatorname{tr}(vv^T)=1$ $A$ is skew-symmetric Is there any nice ...
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### how is the rank of a skew map is even ?

In the book of Linear Algebra by Greub, at page 230, it is claimed that More general, it will now be shown that the rank of a skew transformation is always even. Since every skew mapping is ...
### Rank of a matrix $A$ such that $A + A^T = 0$
I need to prove (using only elementary operations and induction) that rank of a matrix $A\in \Bbb C^{n\times n}$ such that $A + A^T = 0$ is an even number. I know that elementary operations doesn't ...
Consider a convex polygon inscribed in a circle with vertices $P_1$, ..., $P_n, \ n \ge 3$. Let $A$ be the matrix $n \times n$ such that \$ a_{ij} = \begin{cases} |P_iP_j| ...