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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_{m}A and AI_{n} are defined, and I_{m}A = 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