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Which of the following matrices are non-singular?

  1. $I + A$ where $A$ not equal to $0$ is a skew-symmetric real $n\times n$ matrix, $n\geq 2$.
  2. Every skew-symmetric non-zero real $5 \times 5$ matrix.
  3. Every skew-symmetric non-zero real $2 \times 2$ matrix.
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b & c are not true. what about a – poton Aug 7 '12 at 10:47
up vote 2 down vote accepted

We will use the properties $\det A=\det(A^t)$ and $\det(-A)=(-1)^d\det A$, where $d$ is the dimension of the matrix.

  1. We have $\det(A+I)=\det(A^t+I)=\det(I-A)$. Let $x$ such that $Ax=x$. Then $$\langle x,x\rangle=\langle x,Ax\rangle=\langle A^tx,x\rangle=-\langle Ax,x\rangle=-\langle x,x\rangle,$$ which proves that $x=0$, hence $A-I$ is invertible and so is $A+I$.

  2. Using the mentioned properties, for a skew-symmetrix matrix of odd dimension the determiant is $0$.

  3. For a matrix of even dimension, it's not true anymore. Take $A:=\pmatrix{0&1\\-1&0}$, and diagonal blocks of these matrices for general dimension.

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