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ALGEBRAIC PROPERTIES OF MATRIX OPERATIONS

1. Introduction

The properties of matrix operations are useful in solving complex
problems with less e ort. Here we will illustrate several of these
properties via a variety of examples. First, let us point out that the
matrix whose entries are all zeros is called the zero matrix, and is
denoted by O, with the number of rows and columns to be
understood from the context. We begin with some theorems about
matrix operations.

Theorem: Let A, B and C be (m · n) matrices, and let r and s be
real scalars. Then

(1) A + B = B + A
(2) (A + B) + C = A + (B + C)
(3) A + O = A
(4) r(A + B) = rA + rB
(5) (r + s)A = rA + sA
(6) r(sA) = (rs)A

In reading the next theorem, note that I_m is the (m ·m) identity
matrix, which is the (m ·m) matrix such that for each i, the (i, i)
entry is a one and all the other entries are zeros.

Theorem: Let A be an (m · n) matrix.

(1) If B is an (n ·p) matrix and C is a (p ·q) matrix, then the
products A(BC) and (AB)C are defined, and

A(BC) = (AB)C.

(2) If B and C are (n ·p) matrices, then A(B + C) and

AB + AC are de ned and A(B + C) = AB + AC.

(3) If B and C are (k · m) matrices, then (B + C)A and

BA + CA are defined and (B + C)A = BA + CA.

(4) If r is a scalar and B is an (n ·p) matrix, then the products
r(AB), (rA)B and A(rB) are defined and

r(AB) = (rA)B = A(rB).

(5) The products I_mA and AI_n are defined, and I_mA = A = AI_n.

2. Examples of Proofs of Properties of Matrix

Operations

Example: Prove the property (r + s)A = rA + sA.

Proof:

Let r, s ∈ R and let be an (m · n) matrix. Then by the
definition of scalar multiplication, and by the distributive property of
multiplication over addition in R, we have

But by the definitions of scalar multiplication of matrices and matrix
addition, since both rA and sA are (m · n) matrices, we may
compute rA + sA as follows:

Since the corresponding entries of the matrices (r + s)A and rA + sA
are equal, it follows that the matrices (r +s)A and rA+sA are equal:

(r + s)A = rA + sA:

Example: Prove that A(B + C) = AB + AC.

Proof:

Let the columns of B be and let the columns of C be
Then