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Matrix addition and subtraction

You can add and subtract the matrix of the same size as a resulting you get the matrix of the same size.

Definition.
Addition of matrix (sum of matrix) A + B is operation of a finding of a matrix C, entry of which are equal to the sum of all corresponding entry of matrix A and B, can be defined as:

сij = aij + bij

Definition.
Subtraction of matrix (difference of matrix) A - B is operation of a finding of a matrix C, entry of which are equal to a difference of all corresponding entry of matrix A and B, can be defined as:

сij = aij - bij


Properties of matrix addition and subtraction

  • Associativity: (A + B) + C = A + (B + C)
  • A + Θ = Θ + A = A, where Θ - zero matrix
  • A - A = Θ
  • Commutativity: A + B = B + A

Examples of matrix addition and subtraction

Example 1.
Find the sum of matrices A =  (  4  2  )  and B =  (  3  1  ) .
 9  0  -3  4 

Solution:

A + B =  (  4  2  )  +  (  3  1  )  =  (  4 + 3  2 + 1  )  =  (  7  3  )
 9  0   -3  4   9 + (-3)  0 + 4   6  4 
Example 2
Find the difference of matrices A =  (  4  2  )  and B =  (  3  1  ) .
 9  0  -3  4 

Solution:

A - B =  (  4  2  )  -  (  3  1  )  =  (  4 - 3  2 - 1  )  =  (  1  1  )
 9  0   -3  4   9 - (-3)  0 - 4   12  -4 
Example 3
Find the matrix С = 2A + 3B, if A =  (  4  2  )  and B =  (  3  1  ) .
 9  0   -3  4 
 4  -6   9  1 

Solution:

C = 2A + 3B = 2 (  4  2  )  + 3 (  3  1  )  =  (  2·4 + 3·3  2·2 + 3·1  )  =  (  17  7  )
 9  0   -3  4   2·9 + 3·(-3)  2·0 + 3·4   9  12 
 4  -6   9  1   2·4 + 3·9  2·(-6) + 3·1   35  -9 

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