8.1. ADDITION AND SCALAR MULTIPLICATION OF MATRICES 131

Note there are 2×3 zero matrices, 3×4 zero matrices, etc. In fact there is a zero matrixfor every size.

Definition 8.1.7 (Equality of matrices) Let A and B be two matrices. Then A = B meansthat the two matrices are of the same size and for A = (ai j) and B = (bi j) , ai j = bi j for all1≤ i≤ m and 1≤ j ≤ n.

The following properties of matrices can be easily verified. You should do so. Theseproperties are called the vector space axioms.

• Commutative Law Of Addition.

A+B = B+A, (8.1)

• Associative Law for Addition.

(A+B)+C = A+(B+C) , (8.2)

• Existence of an Additive Identity

A+0 = A, (8.3)

• Existence of an Additive Inverse

A+(−A) = 0, (8.4)

Also for α,β scalars, the following additional properties hold.

• Distributive law over Matrix Addition.

α (A+B) = αA+αB, (8.5)

• Distributive law over Scalar Addition

(α +β )A = αA+βA, (8.6)

• Associative law for Scalar Multiplication

α (βA) = αβ (A) , (8.7)

• Rule for Multiplication by 1.1A = A. (8.8)