AcademyMatrix Algebra
Academy
Matrix Addition
Level 1 - Linear Algebra topic page in Matrix Algebra.
Principle
Matrix addition combines matrices entry-by-entry. It is only defined when the matrices have the same size, because every entry in one matrix must have a matching entry in the other.
Notation
\(A,B,C\)
matrices of the same size
\(a_{ij},b_{ij},c_{ij}\)
matching entries in row i and column j
\(A+B\)
the entrywise sum of A and B
\(-A\)
the additive inverse of A
\(0\)
the zero matrix of the required size
The Core Method
If \(A\) and \(B\) are both \(m\) by \(n\), then
Matrix addition
\[(A+B)_{ij}=a_{ij}+b_{ij}\]
Subtract by adding the negative matrix:
Matrix subtraction
\[A-B=A+(-B)\]
For addition and subtraction, check sizes first. If the sizes differ, the operation is not defined.
Matrix addition obeys the usual addition laws when all matrices have the same size:
Commutativity
\[A+B=B+A\]
Associativity
\[(A+B)+C=A+(B+C)\]
Worked Cases
Question
Compute \(\begin{pmatrix}1&-2\\3&0\end{pmatrix}+\begin{pmatrix}4&5\\-1&7\end{pmatrix}\).
Answer
The matrices have the same size, \(2\) by \(2\), so addition is defined. Add matching entries: \(1+4=5\), \(-2+5=3\), \(3+(-1)=2\), and \(0+7=7\). Therefore the sum is \(\begin{pmatrix}5&3\\2&7\end{pmatrix}\).
Examples
Question
Compute \(\begin{pmatrix}6&1\\0&-2\end{pmatrix}-\begin{pmatrix}3&5\\4&-7\end{pmatrix}\).
Answer
Subtract matching entries: \(6-3=3\), \(1-5=-4\), \(0-4=-4\), and \(-2-(-7)=5\). Therefore the difference is \(\begin{pmatrix}3&-4\\-4&5\end{pmatrix}\).