A Direct Proof That a 3 × 3 Skew-Symmetric Matrix Sends Its Defining Vector to Zero

Let v = (p, q, r)T be a vector in ℝ³. From its three components, form the 3 × 3 skew-symmetric matrix

K = (
0−rq
r0−p
−qp0
) .

The entries reflected across the main diagonal have opposite signs, while every diagonal entry is zero. Therefore,

KT = −K.

A Direct Proof That a 3 × 3 Skew-Symmetric Matrix Sends Its Defining Vector to Zero

Multiplication by the vector

Matrix-vector multiplication can be written as a linear combination of the columns of K. The first column is multiplied by p, the second by q, and the third by r:

Kv = p (0r−q) + q (−r0p) + r (q−p0) .

Distributing p, q and r gives

Kv = (
0 · p+(−r) · q+q · r
r · p+0 · q+(−p) · r
−q · p+p · q+0 · r
) .

Combining the entries in each row produces

Kv = (
0 · p + (−r) · q + q · r
r · p + 0 · q + (−p) · r
−q · p + p · q + 0 · r
) .

Since scalar multiplication is commutative, rq = qr, rp = pr and qp = pq. Hence,

Kv = (
−rq + qr
rp − pr
−qp + pq
) = (
0
0
0
) .

Description of the result

The calculation shows directly that the vector (p, q, r)T lies in the null space of its associated skew-symmetric matrix:

K(p, q, r)T = (0, 0, 0)T.

The cancellation is created by the skew-symmetric structure. Every product appears with an equal product of the opposite sign.

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