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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector, \vec\!, adding two matrices would have the geometric effect of applying each matrix transformation separately onto \vec\!, then adding the transformed vectors. :\mathbf\vec + \mathbf\vec = (\mathbf + \mathbf)\vec\!


Definition

Two matrices must have an equal number of rows and columns to be added. In which case, the sum of two matrices A and B will be a matrix which has the same number of rows and columns as A and B. The sum of A and B, denoted , is computed by adding corresponding elements of A and B: :\begin \mathbf+\mathbf & = \begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \cdots & a_ \\ \end + \begin b_ & b_ & \cdots & b_ \\ b_ & b_ & \cdots & b_ \\ \vdots & \vdots & \ddots & \vdots \\ b_ & b_ & \cdots & b_ \\ \end \\ & = \begin a_ + b_ & a_ + b_ & \cdots & a_ + b_ \\ a_ + b_ & a_ + b_ & \cdots & a_ + b_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ + b_ & a_ + b_ & \cdots & a_ + b_ \\ \end \\ \end\,\! Or more concisely (assuming that ): :c_=a_+b_ For example: : \begin 1 & 3 \\ 1 & 0 \\ 1 & 2 \end + \begin 0 & 0 \\ 7 & 5 \\ 2 & 1 \end = \begin 1+0 & 3+0 \\ 1+7 & 0+5 \\ 1+2 & 2+1 \end = \begin 1 & 3 \\ 8 & 5 \\ 3 & 3 \end Similarly, it is also possible to subtract one matrix from another, as long as they have the same dimensions. The difference of A and B, denoted , is computed by subtracting elements of B from corresponding elements of A, and has the same dimensions as A and B. For example: : \begin 1 & 3 \\ 1 & 0 \\ 1 & 2 \end - \begin 0 & 0 \\ 7 & 5 \\ 2 & 1 \end = \begin 1-0 & 3-0 \\ 1-7 & 0-5 \\ 1-2 & 2-1 \end = \begin 1 & 3 \\ -6 & -5 \\ -1 & 1 \end


See also

*
Matrix multiplication In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
* Vector addition * Direct sum of matrices * Kronecker sum


Notes


References

* * {{cite book , last1=Riley , first1=K.F. , last2=Hobson , first2=M.P. , last3=Bence , first3=S.J. , title=Mathematical methods for physics and engineering , edition=3 , publisher=Cambridge University Press , year=2006 , doi=10.1017/CBO9780511810763 , isbn=978-0-521-86153-3 , url=https://archive.org/details/mathematicalmeth00rile , url-access=registration


External links


Matrix Algebra and R
Linear algebra Bilinear maps