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We proceed to develop the algebra of addition of matrices.
Two matrices A and B are said to be conformable for addition if they have the same order (i.e., same number of rows and columns).
If A = (aij)m, n and B = (bij)m, n then their sum A + B is the matrix C = (cij)m,n where cij = aij + bij, i = 1, 2, 3, ...... , m, j = 1, 2, 3, ...., n.
For example:
If A = [a11a12a13a21a22a23a31a32a33] and B = [b11b12b13b21b22b23b31b32b33], then
A + B = [a11+b11a12+b12a13+b13a21+b21a22+b22a23+b23a31+b31a32+b32a33+b33] = C
Note: If A and B be matrices of different orders, then A + B is not defined.
Example on Addition of Matrices:
1. If A = [25β13] and B = [1437], then
A + B = [2+15+4β1+33+7]
= [39210]
2. If A = [β1232β3131β2], B = [3β121032β10] and M = [5214], then
A + B = [β1232β3131β2] + [3β121032β10]
= [β1+32+(β1)3+22+1β3+01+33+21+(β1)β2+0]
= [2153β3450β2]
A + M is not defined since the order of matrix M is not equal to the order of matrix A.
B + M is also not defined since the order of matrix M is not equal to the order of matrix B.
Note: Let A and B are m Γ n matrices and c, d are scalars. Then the following results are obvious.
I. c(A + B) = cA + cB,
For Example:
If A = [2531] and B = [3207] are m Γ n matrices and 5 is scalar. Then
5([2531]+[3207])=5[2531]+5[3207]
II. (c + d)A = cA + dA.
For Example:
If A = [2531] be m Γ n matrix and 5, 3 are scalars. Then
(5+3)[2531]=5[2531]+3[2531]
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