A calculus of the absurd

18.1.3 Properties of matrix addition

For all \(m \times n\) matrices \(A\) and \(B\), there exists a value \(X\) such that

\begin{equation} A + X = B \end{equation}

  • Property 1

    For all values of \(m,n\) there exists an \(m \times n\) matrix \(0\), such that all \(m \times n\) matrices \(A\) satisfy the property

    \begin{equation} A + 0 = 0 + A \end{equation}

  • Property 2 It is also the case that for any \(m \times n\) matrix \(A\), there exists a matrix \(-A\), such that

    \begin{equation} A + (-A) = 0 \end{equation}

  • Property 3 For all \(n \times n\) matrices \(A\) and \(B\), it is the case that

    \begin{equation} A + B = B + A \end{equation}

Together, all these properties mean that \(\mathbb {R}^{m \times n}\) forms an Abelian Group under multiplication.