Adding matrices and scaling them

Adding two matrices: add the entries in the same positions. [1, 4; 0, 2] + [3, –1; 5, 6] = [4, 3; 5, 8]. Subtracting works the same way.

This only works for matrices of the same order: there must be a partner entry for every position. A 2 × 3 and a 3 × 2 matrix cannot be added.

Multiplying by a number k (a scalar): multiply every entry by k. 2[1, –3; 4, 0] = [2, –6; 8, 0].

Addition of matrices behaves like addition of numbers:

• Commutative: A + B = B + A.
• Associative: (A + B) + C = A + (B + C).
• Additive identity: A + O = A.
• Additive inverse: A + (–A) = O, where –A has every sign changed.
• Scalars distribute: k(A + B) = kA + kB and (k + l)A = kA + lA.

Because of these rules, matrix equations are solved just like ordinary equations. From 3X – A = 2B, add A to both sides and divide by 3: X = (2B + A)/3. Then do the arithmetic entry by entry.

Two equations in two unknown matrices work like simultaneous equations. If X + Y = P and X – Y = Q, adding gives 2X = P + Q, and subtracting gives 2Y = P – Q.