Proof That M − Mᵀ Is Skew-Symmetric for a Real 3 × 3 Matrix

This proof demonstrates that subtracting the transpose of any real 3 × 3 matrix from the original matrix produces a 3 × 3 skew-symmetric matrix.

Construction

Let

M =

(

a b c d e f g h i

)

Its transpose is

MT =

(

a d g b e h c f i

)

Subtracting the transpose from the original matrix gives

K = M − MT =

(

0 b − d c − g d − b 0 f − h g − c h − f 0

)

Every diagonal entry is zero because each diagonal entry is subtracted from itself. Entries reflected across the main diagonal have equal magnitudes and opposite signs.

Verification

Taking the transpose of K gives

KT =

(

0 d − b g − c b − d 0 h − f c − g f − h 0

)

Since

d − b = −(b − d),

g − c = −(c − g),

h − f = −(f − h),

it follows that

K^T = −K.

Therefore, K is skew-symmetric. Hence, M − M^T is skew-symmetric for every real 3 × 3 matrix M.