17. Addition of matrices

Addition and subtraction are only defined for matrices of the same type and, indeed, become the addition and subtraction of corresponding elements, respectively. Hence, if

R = (r_ik), S = (s_ik) (i = 1, ··· , m; k = 1, ··· , n)

are two matrices of the type (m,n), then

R + S = (r_ik + s_ik), R - S = (r_ik - s_ik).

Obviously, one has the rules

R + S = S + R (commutative law of addition),
(R + S) + T = R + (S + T) (associative law of addition).

The form of the definition (14.4) of the product shows that the two distributive laws

P(R + S) = PR + PS and (R + S)Q = RQ + SQ

are fulfilled. For the null-matrix O, all elements of which are 0, you have:

R + O = R and RO = OR = O

for every matrix R.

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