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Matrix addition

Define

… in terms of vector addition as the binary operation

\[\begin{align*}
  A + B
  & =
  \begin{pmatrix}
    \vphantom{\bigg(}
    \vec{a}_1 & \cdots & \vec{a}_n
  \end{pmatrix}
  +
  \begin{pmatrix}
    \vphantom{\bigg(}
    \vec{b}_1 & \cdots & \vec{b}_n
  \end{pmatrix}
  \\[1ex]
  & =
  \begin{pmatrix}
    \vphantom{\bigg(}
    \vec{a}_1 + \vec{b}_1 & \cdots & \vec{a}_n + \vec{b}_n
  \end{pmatrix}
\end{align*}
\]

with

where	\(A, B\)	are the terms and
	\(A + B\)	is the sum.

Discuss

• Additive identity

  … is the zero matrix, so

  \[\begin{equation*}
    A + 0_{m \times n} = A,
  \end{equation*}
  \]

• Additive inverse

  … is \(-A\), so

  \[\begin{equation*}
    A + (-A) = 0_{m \times n}.
  \end{equation*}
  \]

• Commutative property

  … holds, so

  \[\begin{equation*}
    A + B = B + A.
  \end{equation*}
  \]

• Associative property

  … holds, so

  \[\begin{equation*}
    (A + B) + C = A + (B + C).
  \end{equation*}
  \]