Matrices
2. Matrices
■ References
• H. Anton and C. Rorres, Elementary Linear Algebra with Supplemental Applications, 12ed
• R. Larson, Elementary Linear Algebra, Metric Version, 8ed
• James Stewart, Calculus, Early Transcendentals, Metric Version
2. Matrices
■ 행렬(Matrix)
Definition 2.1. [Matrix(행렬)]
An m × n matrix on R is of the form


a11 a12 · · · a1n
 a21 a22 · · · a2n 


A= .
..
..
.. 
 ..
.
.
. 
where aij ∈ R
am1 am2 · · · amn
We write
A = aij m × n = aij
or A = aij m × n = aij .
In here,
(1) (A)ij := aij is called the (i, j) component(성분) of A;
(2) ai1 , ai2 , · · · , ain is called the i-th row(행) of A;
(3) a1j , a2j , · · · , amj is called the j-th column(열) of A.
In particular, an n × n matrix is called a square matrix(정사각행렬) of order n.
Example 2.2.
1 × n matrices :
3
1 3 5
2 4 6
2 × 3 matrices :
,
1 3
,
0 2 3
4 1 2
,
1 3 7
m × 1 matrices :
3 × 2 matrices :
Notation 2.3. The set of all m × n matrices on R is denoted by
n
o
Matm × n (R) = [aij ]m × n | aij ∈ R ,
n
o
Matn (R) = Matn × n (R) = [aij ]n × n | aij ∈ R
Definition 2.4. For any A = [aij ], B = [bij ] ∈ Matm × n (R), we define ;
A=B
⇐⇒
aij = bij
for all i, j
Definition 2.5. [Addition and Scalar Multiplication(행렬의 덧셈과 스칼라곱)]
For any A = [aij ], B = [bij ] ∈ Matm × n (R), k ∈ R, we define ;
(1) A + B := [cij ]m × n where cij = aij + bij for all i, j
(2) kA := [kaij ]
(3) O := [cij ] where cij = 0 for all i, j
(4) −A := [−aij ]
(5) A − B = A + (−B)
1

2
,  5 
1




1 3
0 3
 2 4 ,  1 2 
3 6
4 1
3
4

2
Matrices
2. Matrices
Example 2.6.
2 + (−4) 1 + 3 0 + 5
−4 3 5
2 1 0
=
+
(−1) + 2 0 + 2 2 + 0
2 2 0
−1 0 2
2·2 2·1 2·0
2 1 0
=
2A =2
2 · (−1) 2 · 0 2 · 2
−1 0 2
0 0 0
O =O2×3 =
0 0 0
A+B =
Theorem 2.7. For any matrices A, B, C ∈ Matm × n (R) and k, ℓ ∈ R,
A1.
(A + B) + C = A + (B + C)
A2.
A+B =B+A
A3.
A+O =O+A=A
A4.
A + (−A) = (−A) + A = O
SM1.
k(A + B) = kA + kB
SM2.
(k + ℓ)A = kA + ℓA
SM3.
k(ℓA) = (kℓ)A = ℓ(kA)
SM4.
1A = A
Definition 2.8. [Matrix Product(행렬의 곱)]
For any matrices A = [aij ]m × n , B = [bkℓ ]n × r , the product of A and B is the m × r matrix given by
AB := cij m × r ,
where
cij =
n
X
aik bkj = ai1 b1j + ai2 b2j + · · · + ain bnj
k=1

a11 a12
 a21 a22

 ..
..
 .
.
AB = 
 ai1 ai2

 .
..
 ..
.
am1 am2
1 2 4
2 6 0

4
 0
2
···
···
···
···

a1n

a2n 
 b11 b12
..   b
21 b22
. 

 ..
..

ain   .
.
.. 

b
b
n1
n2
.
amn

· · · b1j · · · b1r
· · · b2j · · · b2r 

..
.. 
···
.
··· . 
· · · bnj · · · bnr

4
1 4 3
1 2 4
and B =  0 −1 3 1 . Then
2 6 0
2
7 5 2

1 4 3
∗ ∗ 1·4+2·3+4·5 ∗
12 27 30 13

=
−1 3 1 =
∗ ∗
∗
∗
8 −4 26 12
7 5 2
Example 2.9. Let A =
···
···

(AB)11 = 1 · 4 + 2 · 0 + 4 · 2 = 12
(AB)21 = 2 · 4 + 6 · 0 + 0 · 2 = 8
(AB)12 = 1 · 1 + 2 · (−1) + 4 · 7 = 27
(AB)22 = 2 · 1 + 6 · (−1) + 0 · 7 = −4
(AB)13 = 1 · 4 + 2 · 3 + 4 · 5 = 30
(AB)23 = 2 · 4 + 6 · 3 + 0 · 5 = 26
(AB)14 = 1 · 3 + 2 · 1 + 4 · 2 = 13
(AB)24 = 2 · 3 + 6 · 1 + 0 · 2 = 12
3
Matrices
2. Matrices
Theorem 2.10. For matrices A, B, C, one has ;
(1) A(BC) = (AB)C
(2) A(B + C) = AB + AC and (B + C)A = BA + CA
Remark 2.11. Note that individual column vectors of AB can be obtained by partitioning B into column vectors
and individual row vectors of AB can be obtained by partitioning A into row vectors.
Let A = [aij ] ∈ Matm×n (R) and B = [bst ] ∈ Matn×ℓ (R).
(1) AB computed column by column


b11 b12 · · · b1ℓ
"
#
 b21 b22 · · · b2ℓ 
→
− →
−
→
−


B= .
..
..
..  := b 1 b 2 · · · b ℓ
 ..
.
.
. 
bn1 bn2 · · · bnℓ
"
# "
#
→
− →
−
→
−
→
−
→
−
→
−
AB = A b 1 b 2 · · · b ℓ = A b 1 A b 2 · · · A b ℓ
(2) AB computed row by row

a11 a12 · · ·
 a21 a22 · · ·

A= .
..
..
 ..
.
.

 →

−
a1n
a1
− 
 →
a2n 

 a2 
:=


..
. 
 .. 
. 
→
−
am1 am2 · · · amn
am
 →



−
→
−
a1
a 1B
→
−
→
 a2 
 −



 a 2B 
AB =  .  B = 

.
..
 .. 


→
−
→
−
a
a B
m
m
Definition 2.12. [Linear Combination(일차결합)]
If A1 , A2 , · · · , Ar are matrices of the same size, and if c1 , c2 , · · · , cr are scalars, then an expression of the form
c1 A1 + c2 A2 + · · · + cr Ar
is called a linear combination of A1 , A2 , · · · , Ar with coefficients c1 , c2 , · · · , cr .
−
−
Theorem 2.13. If A is an m × n matrix, and if →
x is an n × 1 column vector, then the product A→
x can be
−
expressed as a linear combination of the column vectors of A in which the coefficients are the entries of →
x.
Proof.

h
−
−
−
A→
x = →
c1 →
c2

x1
i  x2 


−
−
−
−
··· →
c n  .  = x1 →
c 1 + x2 →
c 2 + · · · + xn →
cn
.
 . 
xn
□
4
Matrices
2. Matrices
Definition 2.14. [Transpose Matrix(전치행렬)]
For A = [aij ] ∈ Matm × n (R), the transpose of A is the n × m matrix given by
AT = [bij ]n × m
where bij = aji
Theorem 2.15. For matrices A, B and k ∈ R,
T
T
(1) AT = A
(2) A + B = AT + B T
T
kA = kAT
(3)
(4)
AB
T
= B T AT
Definition 2.16. Let A = [aij ] ∈ Matn (R).
(1) a11 , a22 , · · · , ann are said to be diagonal components(대각성분) of A.
(2) A is called a diagonal matrix(대각행렬) if aij = 0 for all i ̸= j.
1,
(3) A is called the identity matrix(단위행렬, 항등행렬) if aij =
0,
In case, we write I = In .
if i = j,
if i ̸= j
(4) A is called a scalar matrix(상수행렬) if A = kI for some k ∈ R.
(5) A is called an upper triangular matrix(상삼각행렬) if aij = 0 ∀i > j.
A is called a lower triangular matrix(하삼각행렬) if aij = 0 ∀i < j.

a11
 a
 21
 .
 .
 .
an1

k
 0

 .
 .
 .
0
a12
a22
···
an2
0
k
···
0
0
0
..
.
0

a1n
a2n 

.. 

. 
ann
a13
a23
..
.
an3
···
···
···
···

0
0 

.. 
 = kI
. 
k
···
···
···
···

a11
 0

 .
 .
 .
0
0
a22

a11
 0

 .
 .
 .
0
···
···

0
0 

.. 

. 
ann
···
0
0
0
..
.
0
···
···
a12
a22
..
.
0
···
···
..
.
···

a1n
a2n 

.. 

. 
ann

1
 0

 .
 .
 .
0
0
1
···
···
0
a22
..
.
an2
···
···
..
.
···
···
0

a11
 a
 21
 .
 .
 .
an1
···
···
0
0
..
.
0

0
0 

.. 

. 
1
0
0
..
.






ann
■ Invertible Matrix or Nonsingular Matrix(가역행렬 또는 정칙행렬)
Let A = [aij ] ∈ Matn (R).
(1) A is said to be invertible(or non-singular) if there is B ∈ Matn (R) such that AB = BA = I.
In case, we write B = A−1 and A−1 is called the inverse of A. Denote
GL(n, R) = GLn (R) = {A ∈ Matn (R) | A is invertible }
(2) If no such matrix B can be found, then A is said to be a singular matrix(특이행렬).
Example 2.17. In Mat2 (R), for A =
a b
,
c d
If A is invertible, then
A
−1
1
=
ad − bc
d −b
−c
a
A is invertible
⇐⇒
ad − bc ̸= 0.
5
Matrices
2. Matrices
Theorem 2.18. If B and C are inverses of A, then B = C.
Theorem 2.19. For any A, B ∈ GL(n, R) and any k ∈ R with k ̸= 0, In , A−1 , AB, kA ∈ GL(n, R) and
−1
−1
−1
In−1 = In ,
A−1
= A,
AB
= B −1 A−1 ,
kA
= k −1 A−1
■ Power of Matrix(행렬의 거듭제곱)
For A ∈ Matn (R) and m ∈ Z, we define Am inductively as follows
A0 = I
Am+1 = A Am
m
−m
−1
A
= A
m⩾0
if A is invertible and m > 0
m
Remark 2.20. For any A ∈ Matn (R) and m, n ∈ Z, if all of them exist, An Am = An+m , An
= Anm .
Theorem 2.21. Let A ∈ GL(n, R).
−1 m
(1) For any m ∈ Z, Am ∈ GL(n, R) and Am
= A−1 .
−1 T
(2) AT is invertible and AT
= A−1 .
Definition 2.22. [Symmetric Matrix(대칭행렬)]
For A ∈ Matn (R), A is said to be symmetric if AT = A.
Example 2.23.
(1) Every diagonal matrix is symmetric.
(2) For any A ∈ Matm×n (R), AAT and AT A are both symmetric.
Theorem 2.24. Suppose A, B ∈ Matn (R) are symmetric. Then the following matrices are symmetric:
A + B,
kA (k ∈ R),
AT ,
A−1 (if A is invertible.)
Definition 2.25. [Trace(대각합)]
Let A ∈ Matn (R). The trace of A, denoted by trace(A), is defined by
trace(A) =
n
X
aii = a11 + a22 + · · · + ann
i=1
Theorem 2.26. Let A, B ∈ Matn (R) and k ∈ R. Then
trace(A + B) = trace(A) + trace(B),
trace(kA) = k · trace(A),
Theorem 2.27. Let A ∈ Matm×n (R) and B ∈ Matn×m (R). Then
trace(AB) = trace(BA)
trace(AT ) = trace(A)
6
Linear Systems
1. Systems of Linear Equations
1. Systems of Linear Equations
■ Linear Equation(일차방정식)
(1) A linear equation in the n variables x1 , x2 , · · · , xn is of the form
a1 x1 + a2 x2 + · · · + an xn = b
(1–1)
where a1 , a2 , · · · , an and b are constants, and the ai ’s are not all zero.
(2) A solution(해) of the linear equation (1–1) is a sequence of n numbers s1 , · · · , sn such that the equation
(1–1) is satisfied when we substitute x1 = s1 , · · · , xn = sn .
The set of all solution of the equation (1–1) is called its solution set(해집합) or the general solution(일
반해) of the equation.
(3) A linear equation (1–1) is called a homogeneous linear equation(동차일차방정식) where b = 0.
■ Systems of Linear Equations(연립일차방정식)
(1) A finite set of linear equations in the n variables x1 , x2 , · · · , xn is called a system of linear equations
or linear system(선형계). The variables are called unknowns(미지수).
(2) A sequence of n numbers s1 , · · · , sn is called a solution of the linear system if x1 = s1 , · · · , xn = sn is a
solution of every equation in the system.
(3) A system of equations is said to be inconsistent(해가 없다) if it has no solutions; it is called consistent(일
치한다,모순이 없다) if there is at least one solution of the system.
Example 1.1. Linear systems in two unknowns arise in connection with intersections of lines.
ax + by = p
cx + dy = q
line ℓ1
line ℓ2
Example 1.2. Linear systems in three unknowns arise in connection with intersections of planes.

plane α1
 a1 x + b1 y + c1 z = d1
a2 x + b2 y + c2 z = d2
plane α2

a3 x + b3 y + c3 z = d3
plane α3
7
Linear Systems
1. Systems of Linear Equations
■ Augmented Matrix(첨가행렬, 붙임행렬)
Given an arbitrary system of m linear equations in n unknowns can be written as :

a11 x1 + a12 x2 · · · + a1n xn = b1



 a21 x1 + a22 x2 · · · + a2n xn = b2
..

.



am1 x1 + am2 x2 · · · + amn xn = bm

a11
 a21

A=

a12
a22
···
···
..
.
am1 am2 · · · amn


xn
m×n
bm
n×1
m×1
coefficient matrix
unknown vector
constant vector
(계수행렬)
(미지수 벡터)
(상수 벡터)

→
−
−
A→
x = b

b1
 b2 
→
−


b = . 
 .. 

x1
 x2 


→
−
x = . 
 .. 

a1n
a2n 



⇐⇒
h
A
a11

i
→
−
 a21
b =

a12
a22
···
···
..
.
a1n
a2n
am1 am2 · · · amn

b1
b2 

.. 
. 
bm
augmented matrix
■ Homogeneous System of Linear Equations(동차 연립방정식)
→
−
→
−
If b = 0 , then the linear system is said to be homogeneous. Note that every homogeneous linear system
→
−
−
A→
x = 0
has the trivial solution(자명한 해) x1 = x2 = · · · = xn = 0.
Example 1.3. Solve the linear system
2x1 + x2 = 8
x1 + 3x2 = 9
Solution.
(1) ∼ (2)
(2)′ + (1)′ (−2)
(2)′′ ×
−
1
5
(1)′′′ + (2)′′′ (−3)
linear system
2x1 + x2 = 8 · · · · · · (1)
x1 + 3x2 = 9 · · · · · · (2)
x1 + 3x2 = 9 · · · · · · (1)′
2x1 + x2 = 8 · · · · · · (2)′
x1 + 3x2 =
9 · · · · · · (1)′′
− 5x2 = −10 · · · · · · (2)′′
x1 + 3x2 = 9 · · · · · · (1)′′′
x2 = 2 · · · · · · (2)′′′
x1
= 3
x2 = 2
augmented matrix
2 1
8
1 3
9
1 3
9
2 1
8
1
3
9
0 −5
−10
1 3
9
0 1
2
1 0
3
0 1
2
□
8
Elementary Row Operations and Elementary Matrices
1. Systems of Linear Equations
■ Elementary Row Operations(행렬의 기본행연산)
The following operations in Matm×n (R) are called elementary row operations :
• Interchange two rows
• Multiply a row through by a nonzero constant
• Add a multiple of one row to another row
Definition 1.4. [Row Equivalence(행동치)]
Matrices A and B are said to be row equivalent if either (hence each) can be obtained from the other by a
sequence of elementary row operations.
Definition 1.5. [Elementary Matrices(기본행렬=기본행행렬)]
An n × n matrix is called an elementary row matrix if it can be obtained from the n × n identity matrix In by
performing a single elementary row operation.
• Eij : Interchange the i-th row and the j-th row
• Ei (k) : Multiply the i-th row through by k ̸= 0
• Eij (k) : Add the k−multiple of the j-th row to the i-th row


··· 0 ··· 0 ··· 0

.
.
. 
1
· · · .. · · · .. · · · .. 


.
··· 0 ··· 1 ··· 0 
 ..


.
.
.  E (k) =  0
i
· · · .. · · · .. · · · .. 


 ..

··· 1 ··· 0 ··· 0 
 .
..
..
.. 
0
··· . ··· . ··· . 
0 ··· 0 ··· 0 ··· 1
1
 ..
 .

 0


Eij =  ...

 0

 .
 ..


··· 0 ··· 0
.
. 
· · · .. · · · .. 

··· k ··· 0 

..
.. 
··· . ··· . 
··· 0 ··· 1

··· 0 ··· 0 ··· 0
.
.
. 
· · · .. · · · .. · · · .. 

··· 1 ··· k ··· 0 

..
..
.. 
··· . ··· . ··· . 

··· 0 ··· 1 ··· 0 

..
..
.. 
··· . ··· . ··· . 
0 ··· 0 ··· 0 ··· 1
1
 ..
 .

 0


Eij (k) =  ...

 0

 .
 ..
Theorem 1.6 (Row Operations by Matrix Multiplication).
Let A ∈ Matm×n (R). If B results from performing a certain row operation on A and E is the elementary matrix
corresponding the given row operation, then B = EA.
Example 1.7.

(1)

1
0 2 3
 2 −1 3 6  −−→
E23
1
4 4 0

(2)

1
0 2 3
 2 −1 3 6  −−−→
E3 (2)
1
4 4 0

(3)

1 0 2 3
 2 −1 3 6  −−−−−→
E23 (−2)
1 4 4 0
Theorem 1.8. Every elementary matrix is invertible, and the inverse is also an elementary matrix. In fact,
1
−1
,
Eij (k)−1 = Eij (−k)
Eij
= Eij ,
Ei (k)−1 = Ei
k
Remark 1.9. The row equivalence ∼ is an equivalence relation on Matm×n (R).
9
Elementary Row Operations and Elementary Matrices
1. Systems of Linear Equations
■ Reduced Row Echelon Form(기약 행사다리꼴)
Let A ∈ Matm×n (R).
(1) If A has the properties (i)∼(iii), then A is called a matrix in row echelon form(행 사다리꼴).
(2) If A has the properties (i)∼(iv), then A is called a matrix in reduced row echelon form(기약 행사다리
꼴).
(i) If they are any rows that consist entirely of zeros, then they are grouped together at the bottom of
the matrix.
(ii) If a row does not consist entirely of zeros, then the first non-zero number in the row is 1. We call this
a leading 1(선도 1).
(iii) In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs
farther to the right than the leading 1 in the higher row.
(iv) Each column that contains a leading 1 has zeros everywhere else in that column.
Example 1.10.
(1) Every matrix has a unique reduced row echelon form.
(2) Row echelon forms are not unique; that is, different sequences of elementary row operations can result in
different row echelon forms.




1
2 −3
1 2 −3
 2 −1
4  −→ · · · −→  0 1 −2 
1 −1

1
0 0

1


1
2 −3
1 −1
1
 2 −1
4  −→ · · · −→  0 1 −2/3 
1 −1
1
0
0
1
Theorem 1.11. For any A ∈ Matm×n (R), there exist elementary matrices E1 , E2 , · · · , Es and a matrix in reduced
row echelon form R such that
R = Es · · · E1 A
In here, P = Es · · · E1 is an invertible matrix.


1 3 −2 0 1
Example 1.12. Find the reduced row echelon form of  2 6 −3 4 4 .
3 9 −4 8 7
→
−
Theorem 1.13. Let A ∈ Matm×n (R) and b ∈ Matm×1 (R).
→
−
→
−
→
−
→
−
(1) Then
Eib have the same solutions.
h for iany E ∈ GL(m, R), the linear systems A x = b and EA hx =→
−
→
−
−
−
(2) If B →
c is the reduced row echelon form of the augmented matrix A b of the linear system A→
x = b,
→
−
−
−
−
then the linear systems A→
x = b and B →
x =→
c have the same solutions.
Theorem 1.14 (Free Variable Theorem for Homogeneous Systems(동차 연립방정식에 대한 자유변수정리)).
If a homogeneous linear system has n unknowns, and if the reduced row echelon form of its augmented matrix has
r nonzero rows, then the system has n − r free variables(자유변수).
10
Elementary Row Operations and Elementary Matrices
1. Systems of Linear Equations
■ To find the solution of a linear system:
• Gauss-Jordan Elimination(Gauss-Jordan 소거법)
linear system
=⇒ augmented matrix
=⇒ reduced row echelon form
• Gauss Elimination and Back-Substitution(Gauss 소거법과 역대입법)
linear system
=⇒ augmented matrix
=⇒ row echelon form
=⇒ Back-Substitution
Example 1.15. Use Gauss(-Jordan) elimination and back-substitution to solve the linear system
3x1 − 7x2 + 2x3 = 4
2x1 − 4x2 + x3 = 1
Solution.
(1) Gauss-Jordan Elimination:
3 −7 2
2 −4 1
4
1
1 −3 1
2 −4 1
"
1 −3
−→
−→
0
1
3
1
1
− 12
1 −3
1
3
−→
0
2 −1
−5
#
#
"
1
1 0 −2
3
− 29
−→
0 1 − 21
− 52
− 52
By Gauss-Jordan elimination, we have


1
9
1
9
 x1 − 2 x3 = − 2
 x1 = 2 x3 − 2
,


x2 − 12 x3 = − 25
x2 = 12 x3 − 25
Put t = x3 . Then the solutions are
 1   9 

 1


t − 92
−2
2
2
x1





 1
 x2  =  t − 5  = t  1  +  − 5 
2 
 2   2 
 2
x3
t
1
0
(2) Gauss Elimination and Back-Substitution:
3 −7 2
4
1 −3 1
−→
2 −4 1
1
2 −4 1
3
1
(t ∈ R)
−→
1 −3
1
0
2 −1
3
−5
"
−→
1 −3
0
1
1
3
− 12
− 25
By Gauss elimination, we have


 x1 − 3x2 + x3 = 3
 x1 = 3x2 − x3 + 3
,


x2 − 12 x3 = − 25
x2 = 12 x3 − 25
Put t = x3 . Then
x2 = 21 t − 52
x1 = 3x2 − x3 + 3 = 3( 12 t − 25 ) − t + 3 = 12 t − 92
Hence the solutions are
 1

 1   9 
9


−2
2t − 2
2
x1
 1





 x2  =  t − 5  = t  1  +  − 5 
2 
 2
 2   2 
x3
t
1
0
(t ∈ R)
□
#
11
1. Systems of Linear Equations
Elementary Row Operations and Elementary Matrices
Example 1.16. Use Gauss-Jordan elimination to solve the linear system

= 1
 x1 − 3x2 − 2x3
2x1 − 6x2 − 3x3 + 4x4 = 4

3x1 − 9x2 − 4x3 + 8x4 = 7
Example 1.17. Use Gauss-Jordan elimination to solve the linear system

 3x1 + 4x2 − 2x3 = 4
2x1 + 3x2 + 2x3 = 1

4x1 + 5x2 − 6x3 = 8
Solution.
By Gauss-Jordan elimination, we have



3 4 −2
4
1 1 −4


 2 3
2
1 −→ 2 3
2
4 5 −6
8
4 5 −6


3
1 1 −4


1 −→ 0 1 10
8
0 1 10


3
1 1 −4


−5 −→ 0 1 10
−4
0 0
0
Hence the given linear system has no solution, i.e., it is inconsistent.

3
−5 
1
□
Theorem 1.18. If R is the reduced row echelon form of an n × n matrix A, then either R = In or R has a row
of zeros.
Theorem 1.19. For a matrix A ∈ Matn (R), the following are equivalent :
(1) A is invertible.
→
−
−
(2) A→
x = 0 has only the trivial solution.
(3) The reduced row echelon form of A is In .
(4) A is expressible as a product of elementary matrices.
■ Inversion Algorithm(역행렬 알고리즘)
To find the inverse of an invertible matrix A, find a sequence of elementary row operations that reduces A to the
identity and then perform that same sequence of operations on In to obtain A−1 . That is, if A ∈ Matn (R) is a
invertible matrix, by the Theorem 1.19, there exist elementary matrices E1 , · · · , Es such that
Es · · · E1 A = In ,
A−1 = Es · · · E1
Thus
A −→ E1 A −→ E2 E1 A −→ · · · −→ Es · · · E1 A = In = R
In −→ E1
−→ E2 E1
−→ · · · −→ Es · · · E1 = A−1
Hence
h
A
i
h
I −→ E1 A
i
h
E1 −→ · · · −→ Es · · · E1 A
i h
Es · · · E1 = I
A−1
i
12
Invertible matrices
1. Systems of Linear Equations

1 0 1
Example 1.20. Use the inversion algorithm to find the inverse of A =  1 1 3 .
2 5 13

Theorem 1.21. A system of linear equations has zero, one, or infinitely many solutions. There are no other
possibilities.
→
−
→
−
−
Theorem 1.22. If A is invertible, then for each n × 1 matrix b , the linear system A→
x = b has exactly one
→
−
−
solution, namely →
x = A−1 b .
Theorem 1.23. Let A ∈ Matn (R).
(1) If there is B ∈ Matn (R) such that BA = I, then A is invertible and B = A−1 .
(2) If there is B ∈ Matn (R) such that AB = I, then A is invertible and B = A−1 .
Theorem 1.24. If AB is invertible, then A and B are both invertible.
Theorem 1.25. For a matrix A ∈ Matn (R), the following are equivalent :
(1) A is invertible.
→
−
−
(2) A→
x = 0 has only the trivial solution.
(3) The reduced row echelon form of A is In .
(4) A is expressible as a product of elementary matrices.
→
−
→
−
−
(5) The linear system A→
x = b has exactly one solution for every n × 1 matrix b .
→
−
→
−
−
(6) The linear system A→
x = b is consistent for every n × 1 matrix b .
Proof.


0
 .. 
 . 
 
→
−
→
−
→
−

(6) =⇒ (1) For each e k = 
 1  (k = 1, · · · , n), the linear system A x = e k has a solution.
 .. 
 . 
0
−
−
−
Let →
x 1, →
x 2, · · · , →
x n be solutions of the respective systems. Then
h
i h
i h
i
−
−
−
−
−
−
−
−
−
A →
x →
x ··· →
x = A→
x A→
x · · · A→
x = →
e →
e ··· →
e
=I
1
2
n
1
2
h
i
−
−
−
x2···→
x n is the inverse of A.
Hence by Theorem 1.24, →
x 1→
n
1
2
n
n
□