[Math] Skew Symmetric Matrix Properties

block matriceslinear algebramatricesmatrix decompositionmatrix equations

We have a theorem says that "ODD-SIZED SKEW-SYMMETRIC MATRICES ARE
SINGULAR" . Proof link is given here if needed. Now let us assume we have a $3\times 3$ skew symmetric matrices of the form $ \begin{bmatrix}\,0&\!-a_3&\,\,a_2\\ \,\,a_3&0&\!-a_1\\-a_2&\,\,a_1&\,0\end{bmatrix}$ and an Identity matrix $I_{3\times3}$

Question

  1. Can we say determinant of $I_{3\times3}+\begin{bmatrix}\,0&\!-a_3&\,\,a_2\\ \,\,a_3&0&\!-a_1\\-a_2&\,\,a_1&\,0\end{bmatrix} \tag 1$

    is not zero always? if so how can we prove it mathematically ?

NB:: $a_1,a_2,a_3$ cant be zero together at a time

Best Answer

Another way: if $A$ is skew-symmetric, then for any vector $v$ we have $v^T A v = - v^T A^T v = - v^T A v$ so $v^T A v = 0$. Now for any real vector $v$, if $(I + A) v = 0$ we have $$0 = v^T (I+A) v = v^T v + v^T A v = v^T v$$ so $v = 0$. This works whether $n$ is odd or even.

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