Matrices: Definition, Addition, Multiplication, and Properties, Study Guides, Projects, Research of Linear Algebra

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MTHSC 412 Section 1.6 – Matrices

Kevin James

Matrices

Definition

1 Let S be a set. An m × n matrix over S will be an array

A =

a 1 , 1 a 1 , 2... a 1 ,n a 2 , 1 a 2 , 2... a 2 ,n .. .

am, 1 am, 2... am,n

with m rows and n columns and elements ai,j ∈ S. 2 Two matrices A and B over S are equal if they have the same dimensions and if ai,j = bi,j for 1 ≤ i ≤ m and 1 ≤ j ≤ n. 3 We denote the set of m × n matrices over S by Mm×n(S). 4 We denote the set of n × n (square) matrices over S by Mn(S).

Matrix Multiplication for Matrices over R

Definition Suppose that A ∈ Mm×n(R) and B ∈ Mn×p (R). Then we define the product AB to be the m × p matrix C = AB with

ci,j =

∑^ n

k=

ai,k bk,j.

Note

Matrix multiplication encodes composition of functions. Let A ∈ Mm×n(R) and B ∈ Mn×p (R). Define functions f : Rn^ → Rm^ and g : Rp^ → Rn^ by

f (~v ) = A~v and g (~w ) = B w~.

Then f ◦ g : Rp^ → Rm^ is a linear map with matrix C = AB That is, f ◦ g (~v ) = C~v where C = AB.

Or put another way, matrix multiplication is defined so that

A(B~v ) = (AB)~v.

Theorem

Matrix multiplication is associative. That is if A ∈ Mm×n(R), B ∈ Mn×p (R) and C ∈ Mp×r (R), then (AB)C = A(BC ).

Note

This follows from our recognition of matrix multiplication as composition of functions and our proof that compositions of functions is associative.

Proof.

First we note that AB is m × p and thus (AB)C is m × r. Also, BC is n × r and thus A(BC ) is m × r. Thus A(BC ) and (AB)C both have the same dimensions. Also, for 1 ≤ i ≤ m and 1 ≤ j ≤ r , we have

[A(BC )]i,j =

∑^ n

k=

ai,k [BC ]k,j =

∑^ n

k=

ai,k

∑p

s=

bk,s cs,j

∑^ p

s=

∑^ n

k=

ai,k (bk,s cs,j ) =

∑^ p

s=

∑^ n

k=

(ai,k bk,s )cs,j

∑^ p

s=

cs,j

∑n

k=

ai,k bk,s =

∑^ p

s=

cs,j [AB]i,s

∑^ p

s=

[AB]i,s cs,j = [(AB)C ]i,j

Thus A(BC ) = (AB)C.

A special matrix and its properties

Definition

Define In ∈ Mn(R) by

[In]i,j =

1 if i = j, 0 otherwise.

Theorem

Suppose that A ∈ Mm×n(R). Then,

1 ImA = A. 2 AIn = A.

Left and Right Inverses

Definition

Let ∗ be a binary operation on a nonempty set A.

1 If e ∈ A satisfies e ∗ a = a for all a ∈ A, then e is said to be a left identity for A with respect to ∗. 2 If e ∈ A satisfies a ∗ e = a for all a ∈ A, then e is said to be a right identity for A with respect to ∗. 3 Note that if e ∈ A is both a left and right identity then it is an identity.

Square Matrices

Theorem

1 Addition is an associative binary operation on Mn(R). 2 The zero matrix is an identity for Mn(R) w. r. t. addition. 3 Each matrix A ∈ Mn(R) has an inverse w. r. t. addition , namely −A. 4 Addition is commutative. 5 Multiplication is an associative binary operation on Mn(R). 6 In is an identity for Mn(R) w. r. t. matrix multiplication. 7 A ∈ Mn(R) has an inverse w. r. t. matrix multiplication if and only if det(A) 6 = 0 (from linear algebra). 8 For A, B, C ∈ Mn(R), we have A(B + C ) = AB + AC and (A + B)C = AC + BC.