Deriving the Outer-Product Matrix BAᵀ from Matrix Multiplication

Let A, B and X be column vectors in ℝ³:

A =
a₁
a₂
a₃
, B =
b₁
b₂
b₃
, X =
x
y
z
.

BAᵀX = B(AᵀX).

Matrix multiplication is associative, so AᵀX may be evaluated first. Since Aᵀ is a 1 × 3 row matrix and X is a 3 × 1 column matrix, their product is a scalar:

AᵀX = a₁x + a₂y + a₃z.

Therefore,

BAᵀX =
b₁
b₂
b₃
(a₁x + a₂y + a₃z).

The quantity in parentheses is a scalar, so it multiplies every component of B:

BAᵀX =
b₁a₁x + b₁a₂y + b₁a₃z
b₂a₁x + b₂a₂y + b₂a₃z
b₃a₁x + b₃a₂y + b₃a₃z
.

Collect the coefficients of x, y and z into a matrix multiplying X:

BAᵀX =
b₁a₁b₁a₂b₁a₃
b₂a₁b₂a₂b₂a₃
b₃a₁b₃a₂b₃a₃
x
y
z
.

Since this holds for every vector X, the matrix multiplying X must be BAᵀ. Hence,

BAᵀ =
b₁a₁b₁a₂b₁a₃
b₂a₁b₂a₂b₂a₃
b₃a₁b₃a₂b₃a₃
.

The complete derivation is compressed into the identity

BAᵀX = B(A · X).

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