2.4 Properties of Operations on
Linear Transformations and
Matrices
Goal: Show that matrix operations enjoy many (but
not all!!!) of the properties of the analogous operations
on ordinary real numbers.
Section 2.4 Properties of Operations on Linear Transformations and Matrices
1
Properties of Matrix Addition and
Scalar Multiplication
Theorem: If A, B and C are m  n matrices, and r and
s are scalars, then the following properties hold:
1. The Commutative Property of Addition:
AB  BA
2. The Associative Property of Addition:
A  B  C   A  B   C
3. The “Left” Distributive Property:
r  s A  rA  sA
4. The “Right” Distributive Property:
rA  B   rA  rB
5. The Associative Property of Scalar Multiplication:
rsA   rs A  srA 
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Section 2.4 Properties of Operations on Linear Transformations and Matrices
Properties of Matrix Multiplication
Theorem: If A and B are m  k matrices, C and D are
k  n matrices, and r is a scalar, then the following
properties hold:
1. The “Left” Distributive Property:
A  B C  AC  BC
2. The “Right” Distributive Property:
AC  D   AC  AD
3. The Associative Property of Mixed (Scalar and
Matrix) Products:
rBC   rB C  BrC 
Section 2.4 Properties of Operations on Linear Transformations and Matrices
3
The Associative Property of Matrix
Multiplication
Theorem: If A is an m  p matrix, B is a p  q matrix,
and C is a q  n matrix, then ABC   AB C.
Proof:
Both products ABC  and AB C are m  n matrices.
Now, we have to show that both sides, pair-wise, have
exactly the same entries.
Case 1: C  x, a q  1 matrix.
B 
b1 
b2 … 
bq
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Section 2.4 Properties of Operations on Linear Transformations and Matrices
 1 Ab
 2 … Ab
q
AB  Ab
x1
 1 Ab
 2 … Ab
q
AB x  Ab
x2

xq
 1  x 2 Ab
 2    x q Ab
q
 x 1 Ab
Now, let us work on ABx:
x1
Bx  
b1 
b2 … 
bq
x2

xq
 x 1
b 1  x 2
b 2    x q
bq
Section 2.4 Properties of Operations on Linear Transformations and Matrices
5
ABx  A x 1 
b 1  x 2
b 2    x q
bq
 A x 1
b 1  A x 2
b 2    A x q
bq
by the “Right” Distributive Property)
 1  x 2 Ab
 2    x q Ab
q
 x 1 Ab
Case 2: C is an arbitrary q  n matrix:
C  c 1 c 2 ... c n
AB c i  ABci 
for every column c i .
Thus, column i of AB C is exactly the same as that of
ABC , and therefore AB C  ABC .
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Section 2.4 Properties of Operations on Linear Transformations and Matrices
The Matrix of a Composition
Theorem: If T 1 :  n →  k and T 2 :  k →  m are
linear transformations, then:
T 2 ∘ T 1   T 2 T 1 
Section 2.4 Properties of Operations on Linear Transformations and Matrices
7
k −fold Compositions
If T 1 , T 2 , . . . . , T k−1 , T k are all linear transformations
with the property that the codomain of T i is the
domain of T i1 , for all i  1. . k − 1, then we can
inductively construct the k −fold composition of these
linear transformations by:
T k ∘ T k−1 ∘  ∘ T 2 ∘ T 1 v 
 T k T k−1 ∘  ∘ T 2 ∘ T 1 v 
T k ∘ T k−1 ∘  ∘ T 2 ∘ T 1   T k T k−1 T 2 T 1 
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Section 2.4 Properties of Operations on Linear Transformations and Matrices
Powers of Square Matrices and
Linear Operators
Theorem: The matrix product AA can be formed if
and only if A is an n  n matrix. Analogously, the
composition T ∘ T can be formed if and only if
T :  n →  n , i.e., T is an operator.
Write AA as A 2 and T ∘ T as T 2 .
Similarly, by induction, we will write:
A k  A  A k−1  A  A    A, and
T k v   TT k−1 v   TT. . . Tv 
Section 2.4 Properties of Operations on Linear Transformations and Matrices
9
Evaluating a Polynomial with a
Matrix:
Definition: If px   c 0  c 1 x  c 2 x 2    c k x k is a
polynomial with real coefficients, and A is any n  n
matrix, then we define the polynomial evaluation,
pA , by:
pA   c 0 I n  c 1 A  c 2 A 2    c k A k .
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Section 2.4 Properties of Operations on Linear Transformations and Matrices
Multiplication by I n and 0 mn
Theorem: If A is any m  n matrix, then:
AI n  A
and
I m A  A;
A0 np  0 mp
and
0 qm A  0 qn .
Section 2.4 Properties of Operations on Linear Transformations and Matrices
11
Danger Zone!
The Existence of Zero Divisors:
Definition: Two n  n matrices A and B with the
property that AB  0 nn , but neither A nor B is 0 nn
are called zero divisors.
In other words, The Zero Factors Theorem does not
hold for matrices.
AB ≠ BA Most of the Time!
Matrix multiplication,
commutative!
12
in
general,
is
NOT
Section 2.4 Properties of Operations on Linear Transformations and Matrices
T is Uniquely Determined by a
Basis
Theorem: If T :  n →  m is a linear transformation,
and B  v 1 , v 2 , . . . , v n  is any basis for  n , then the
action of T is uniquely determined by the vectors
Tv 1 , Tv 2 , . . . , Tv n  from  m . More specifically,
if v ∈  n and v is expressed (uniquely) as
v  c 1v 1  c 2v 2    c nv n , then:
Tv   c 1 Tv 1   c 2 Tv 2     c n Tv n .
Section 2.4 Properties of Operations on Linear Transformations and Matrices
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