Properties of Matrix Operations
2010, 14, Sep.
Ki-Seung Lee
Properties of Matrix Addition
A+B = B+A
A+(B+C)=(A+B)+C
(cd)A = c(dA)
1A=A
c(A+B) = cA+cB
(c+d)A = cA +dA
A+0mn=A
A+(-A) = 0mn
If cA=0mn then c=0 or A=0mn.
Commutative
Associative
Scalar Associative
Scalar identity
Scalar distributive 1
Scalar distributive 2
Additive identity
Additive Inverse
Scalar cancellation property
Properties of Matrix Multiplication
A(BC) = (AB)C
A(B+C) = AB +AC
(A+B)C = AC+BC
c(AB) = (cA)B=A(cB)
AIn = A
ImA = A
assuming A is m by n and a
ll operations are defined.
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–
–
–
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Associative
Left distributive
Right Distributive
Scalar Associative
Multiplicative Identity
Multiplicative Identity
Using Properties to Prove Theorems
• Using these properties we can prove the following the
orem (which we have already been assuming).
• Theorem: For a system of linear equations in
n variables, precisely one of the following is tr
ue:
1. The system has exactly one solution.
2. The system has an infinite number of solutions.
3. The system has no solutions.
The Transpose of a Matrix
• We will find it useful at times to talk about the
transpose of a matrix.
• Given an m by n matrix A, we define AT (A tran
spose) to be the n by m matrix:
a1,1
a1,2
A T  

a1,n
a2,1
am ,1 
a2,2
am,2 
.


am,n 
a2,n
Properties of Transposes
1. (AT)T = A
Transpose of a transpose
2. (A + B) T = AT+BT
Transpose of a sum
3. (cA)T = c(AT)
Transpose of a scalar prod
uct
Transpose of a product
4. (AB)T = BTAT
What about Mult. Inverses
• For an n by n matrix A, can we find an n by n
matrix A-1 so that
AA-1=A-1A=In ?
• Does this always work?
Properties of Inverse Matrices
Definition
• Last time we said the the inverse of an n by n
matrix A is an n by n matrix B where,
AB = BA = In.
• We also talked about how to find the inverse o
f a matrix and said that not all matrices have in
verses (some are singular) so won’t review that
here.
Properties of Inverses
1. If A is an invertible matrix then its inverse is u
nique.
2. (A-1)-1 = A.
3. (Ak)-1= (A-1)k (we will denote this as A-k )
4. (cA)-1 = (1/c)A-1, c ≠ 0.
5. ( AT)-1 = (A-1)T.
Some theorems involving Inverses
1. If A and B are invertible matrices then,
(AB)-1 = B-1A-1.
2. If C is an invertible matrix then the following prop
erties hold.
a) If AC = BC then A = B.
b) If CA = CB then A = B.
3. If A is an invertible matrix, then the system of equ
ations Ax = b has a unique solution given by
x = A-1b.
Elementary Matrices
• An n by n matrix is called an elementary matr
ix if it can be obtained from In by a single elem
entary row operation.
• These matrices allow us to do row operations
with matrix multiplication.
Representing Elementary Row Operations
Theorem: Let E be the elementary matrix obtain
ed by performing an elementary row operation
on In. If that same row operation is performed
on an m by n matrix A, then the resulting matri
x is given by the product EA.
Row equivalent matrices
• Let A and B be m by n matrices. Matrix B is ro
w equivalent to A if there exists a finite numbe
r of elementary matrices E1, E2, ... Ek such that
B = EkEk-1 . . . E2E1A.
LU Factorizations
Review of Last Time
• An elementary matrix is a matrix that can be o
btained from an identity matrix by applying a s
ingle row operation
• The inverse of an elementary matrix is also an
elementary matrix
• Doing row operations can be seen as multiplyi
ng by an elementary matrix
Fact from last time
• A square matrix A is row equivalent to the iden
tity matrix if and only if it can be written as a p
roduct of elementary matrices.
• Theorem: A square matrix A is invertible if an
d only if it can be written as the product if ele
mentary matrices.
The BIG Theorem
The following statements are equivalent (TFS
AE) for any n by n square matrix A.
1. A is invertible.
2. Ax = b has a unique solution for any n by 1 matrix
b.
3. Ax = 0 has only the trivial solution.
4. A is row-equivalent to In.
5. A can be written as the product of elementary matr
ices.
A few obvious definitions
• A matrix A is said to be upper triangular if ak
,,l  0 implies l ≥ k.
• A matrix A is said to be lower triangular if ak,
,l  0 implies k ≥ l.
Why do we care???
• If we could somehow factor an n by n matrix A
into a lower triangular matrix L and an upper tr
iangular matrix U,
A = LU
then we can solve any system of equations Ax
= b without doing row operations.
So how can we find LU factorizations??
• Let’s start by trying to row reduce A to an uppe
r triangular matrix.
• What could happen to make this not work?