Matrices and Determinants
Advanced Level Pure Mathematics
Advanced Level Pure Mathematics
8
Chapter 8
Matrices and Determinants
8.1
INTRODUCTION : MATRIX / MATRICES
2
8.2
SOME SPECIAL MATRIX
3
8.3
ARITHMETRICS OF MATRICES
4
8.4
INVERSE OF A SQUARE MATRIX
16
8.5
DETERMINANTS
19
8.6
PROPERTIES OF DETERMINANTS
21
8.7
INVERSE OF SQUARE MATRIX BY DETERMINANTS
27
Prepared by K. F. Ngai
Page 1
Matrices and Determinants
Advanced Level Pure Mathematics
8.1
INTRODUCTION : MATRIX / MATRICES
1. A rectangular array of mn numbers arranged in the form
 a11 a12  a1n 
 a 21 a 22  a 2 n 
 
 


a m1 a m2  a mn 
is called an mn matrix.
 2 3 4
e.g. 
is a 23 matrix.
 1 8 5
2
e.g.  7  is a 31 matrix.
 3
 
2. If a matrix has m rows and n columns, it is said to be order mn.
 2 0 3 6
e.g.  3 4 7 0 is a matrix of order 34.
 1 9 2 5


 1 0 2
e.g.  2 1 5  is a matrix of order 3.
 1 3 0 


3.
a
4.
 b1 
b2 
   is called a column matrix or column vector.
 
bn 
1
a 2  a n  is called a row matrix or row vector.
2
e.g.  7  is a column vector of order 31.
 3
 
Prepared by K. F. Ngai
Page 2
Matrices and Determinants
e.g.  2 3 4 is a row vector of order 13.
Advanced Level Pure Mathematics
5. If all elements are real, the matrix is called a real matrix.
 a11 a12  a1n 
a
a 22  a 2 n 
6.  21
is called a square matrix of order n.

  


a n1 a n 2  a nn 
And a11 , a 22 ,  , a nn is called the principal diagonal.
3 9 
e.g. 
is a square matrix of order 2.
 0 2
 
7. Notation : a ij
8.2
m n
a 
,
ij
m n
, A , ...
SOME SPECIAL MATRIX.
Def.8.1 If all the elements are zero, the matrix is called a zero matrix or null matrix,
denoted by Om n .
 0 0
e.g. 
is a 22 zero matrix, and denoted by O2 .
 0 0
 
Def.8.2 Let A  a ij
n n
be a square matrix.
(i) If aij  0 for all i, j, then A is called a zero matrix.
(ii) If aij  0 for all i<j, then A is called a lower triangular matrix.
(iii) If aij  0 for all i>j, then A is called a upper triangular matrix.
 a11
 a 21
 

 a n1
i.e.
0
a 22
an2
 a11 a12  a1n 
 0 a 22
 
0
0
 


 

 0  0 a nn 
Upper triangular matrix
0
 
0

  a nn 
0 
0
Lower triangular matrix
 1 0 0
e.g.  2 1 0 is a lower triangular matrix.
 1 0 4 


Prepared by K. F. Ngai
Page 3
Matrices and Determinants
Advanced Level Pure Mathematics
 2 3
e.g. 
is an upper triangular matrix.
 0 5 
 
Def.8.3 Let A  a ij
n n
be a square matrix. If aij  0 for all i  j , then A is called a
diagonal matrix.
1 0 0
e.g.  0 3 0
 0 0 4


is a diagonal matrix.
Def.8.4 If A is a diagonal matrix and a11  a 22   a nn  1 , then A is called an
identity matrix or a unit matrix, denoted by I n .
1 0 0
I 3   0 1 0
 0 0 1


e.g.
 1 0
I2  
,
 0 1
8.3
ARITHMETRICS OF MATRICES.
Def.8.5 Two matrices A and B are equal iff they are of the same order and their
corresponding elements are equal.
i.e.
a 
ij
m n
 
 bij
m n
e.g.
 a 2 1 c
 4 b   d 1
N.B.
 2 3  2 4
 4 0   3 0
 
Def.8.6 Let A  a ij
m n


and
aij  bij for all i , j .
a  1, b  1, c  2, d  4 .
 2 1
 3 0  2 3 1
1 4   1 0 4 


 
and B  bij
m n
.
 
Define A  B as the matrix C  cij
m n
of the same order such that
cij  aij  bij for all i=1,2,...,m and j=1,2,...,n.
e.g.
2 3 1 2 4 3
 1 0 4   2 1 5 
Prepared by K. F. Ngai
Page 4
Matrices and Determinants
Advanced Level Pure Mathematics
 2 1
2 3 1
N.B.
1.  3 0  
is not defined.
1 4  1 0 4 


2 3
 5 is not defined.
2. 
4 0
Def.8.7 Let A  aij m n . Then  A  aij m n and A-B=A+(-B)
 
e.g.1
 
 1 2 3
2 4 0
B

If A  
and
 3 1 1 . Find -A and A-B.
1 0 2
Thm.8.1
Properties of Matrix Addition.
Let A, B, C be matrices of the same order and O be the zero matrix of the
same order. Then
(a) A+B=B+A
(b) (A+B)+C=A+(B+C)
(c) A+(-A)=(-A)+A=O
(d) A+O=O+A
Def.8.8
Scalar Multiplication.
Let A  aij m n , k is scalar. Then kA is the matrix C  cij
 
 
by cij  kaij ,
 
i.e. kA  kaij
e.g.
defined
i, j.
m n
 3 2
If A  
,
5 6 
then
N.B.
m n
-2A=
;
3
A
2
(1) -A=(-1)A
(2) A-B=A+(-1)B
Prepared by K. F. Ngai
Page 5
Matrices and Determinants
Advanced Level Pure Mathematics
Thm.8.2
Properties of Scalar Multiplication.
Let A, B be matrices of the same order and h, k be two scalars.
Then (a) k(A+B)=kA+kB
(b) (k+h)A=kA+hA
(c) (hk)A=h(kA)=k(hA)
Let A  aij m n . The transpose of A, denoted by A T , or A  , is
 
Def.8.9
defined by
a11
a
A T   12


a1n
e.g.
 3 2
A
,
5 6 
e.g.
 3 0 2
A
,
4 6 1 
e.g.
A  5 ,
N.B.
a 21  a m1 
a 22  a m2 
 

a 2 n  a nm  n  m
then A T 
then A T 
then A T 
(1) I T 
(2) A  aij
 
m n
,
then A T 
Thm.8.3
Properties of Transpose.
Let A, B be two mn matrices and k be a scalar, then
(a) ( A T ) T 
(b) ( A  B ) T 
(c) (kA ) T 
Def.8.11
A square matrix A is called a symmetric matrix iff A T  A .
i.e.
e.g.
e.g.
Def.8.12
A is symmetric matrix  A T  A  aij  a ji i, j
 1 3 1
 3 3 0 
 1 0 6 


 1 3 1
 0 3 0 
 1 3 6 


is a symmetric matrix.
is not a symmetric matrix.
A square matrix A is called a skew-symmetric matrix iff A T   A .
Prepared by K. F. Ngai
Page 6
Matrices and Determinants
Advanced Level Pure Mathematics
i.e. A is skew-symmetric matrix  A T   A  aij  a ji i, j
e.g.2
 0 3 1
Prove that A  3 0 5  is a skew-symmetric matrix.
 1 5 0 


e.g.3
Is a ii  0 for all i=1,2,...,n for a skew-symmetric matrix?
Def.12 Matrix Multiplication.
Let A  aik mn and B  bkj
 
matrix C  cij
 
m p
n p
. Then the product AB is defined as the mp
where
n
cij  ai 1b1 j  ai 2 b2 j ain bnj   aik bkj .
k 1
i.e.
n
AB   aik bkj 
 k 1
 m p
Prepared by K. F. Ngai
Page 7
Matrices and Determinants
Advanced Level Pure Mathematics
e.g.4
 2 1
 2 3 1 . Find AB and BA.
Let A   3 0
and B  
 1 4 
 1 0 4  2 3

 3 2
e.g.5
 2 1
Let A   3 0 and B 
1 4

 3 2
N.B.
Thm.8.4
 1 0 . Find AB. Is BA well defined?
2 1 2  2
In general, AB  BA .
i.e. matrix multiplication is not commutative.
Properties of Matrix Multiplication.
(a) (AB)C = A(BC)
(b) A(B+C) = AB+AC
(c) (A+B)C = AC+BC
(d) AO = OA = O
(e) IA = AI = A
(f) k(AB) = (kA)B = A(kB)
Prepared by K. F. Ngai
Page 8
Matrices and Determinants
Advanced Level Pure Mathematics
(g) ( AB)  B A .
T
N.B.
T
T
(1) Since AB  BA ;
Hence, A(B+C)  (B+C)A
and
A(kB)  (kB)A.
(2) A 2  kA  A( A  kI )  ( A  kI ) A .
(3) AB  AC  O  A(B  C)  O

 A  O o rB  C  O
1 0
 0 0
 0 0
e.g. Let A  
, B  
, C  

 0 0
 0 1
 1 0
1 0  0 0  1 0  0 0
Then AB  AC  

 


 0 0  0 1  0 0  1 0
 0 0  0 0  0 0

 
 

 0 0  0 0  0 0
But
so
Def.
AO
and
B  C,
AB  AC  O 
 A  O or B  C .
Powers of matrices
For any square matrix A and any positive integer n, the symbol
A n denotes 
A 
A
 A

A .


n factors
N.B.
e.g.6
(1) ( A  B ) 2  ( A  B )( A  B )
 AA  AB  BA  BB
 A 2  AB  BA  B 2
(2) If AB  BA , then ( A  B ) 2  A 2  2 AB  B 2
 2 1
 1
1 2 3
2 4 0




Let A  
, B
 , C  1 0 and D   2
 1 0 2 
 3 1 1


 
 1 1
 0
Evaluate the following :
(a) ( A  2B)C
(b) ( AC ) 2
(c) ( B T  3C ) T D
(d) ( 2 A) T B  DD T
Prepared by K. F. Ngai
Page 9
Matrices and Determinants
Advanced Level Pure Mathematics
Prepared by K. F. Ngai
Page 10
Matrices and Determinants
Advanced Level Pure Mathematics
e.g.7
(a) Find a 2x2 matrix A such that
 1 0 1 
 1 0  .
2 A  3
  A  

 1 1 2 
 1 1 
2
(b) Find a 2x2 matrix A  

2 1
A T  A and 
A
 3 0

 such that

 2 1 .
A

 3 0
3 1  1   0  1
(c) If 
   
   , find the values of x and  .
 1 1   x   0    x
Prepared by K. F. Ngai
Page 11
Matrices and Determinants
Advanced Level Pure Mathematics
e.g.8
e.g.9
cos  sin  
Let A  
 . Prove by mathematical induction that
 sin  cos 
 cos n  sin n 
[HKAL92] (3 marks)
An  
 for n = 1,2, .
 sin n cos n 
(a) Let A  
a 1
 where a , b  R and a  b .
 0 b
 n
Prove that A n   a

 0
a n  bn 

a  b  for all positive integers n.
bn 
95
1 2
(b) Hence, or otherwise, evaluate 
 .
 0 3
[HKAL95] (6 marks)
Prepared by K. F. Ngai
Page 12
Matrices and Determinants
Advanced Level Pure Mathematics
e.g.10
 0 1 0
(a) Let A   0 0 1 and B be a square matrix of order 3. Show that if A


 0 0 0
and B are commutative, then B is a triangular matrix.
(b) Let A be a square matrix of order 3. If for any x , y , z  R , there exists
 x
 x


  R such that A y    y , show that A is a diagonal matrix.
 
 
 z
 z
(c) If A is a symmetric matrix of order 3 and A is nilpotent of order 2 (i.e.
then A=O, where O is the zero matrix of order 3.
A 2  O ),
Prepared by K. F. Ngai
Page 13
Matrices and Determinants
Advanced Level Pure Mathematics
Properties of power of matrices :
(1) Let A be a square matrix, then ( A n ) T  ( A T ) n .
(2) If AB  BA ,
then
(a) ( A  B) n  A n  C1n A n1 B  C2n A n 2 B 2  C3n A n 3 B 3  Cnn1 AB n1  B n
(b) ( AB ) n  A n B n .
(3) ( A  I ) n  A n  C1n A n1  C2n A n 2  C3n A n 3  Cnn1 A  Cnn I
e.g.11 (a) Let X and Y be two square matrices such that XY = YX.
(i) ( X  Y ) 2  X 2  2 XY  Y 2
Prove that
n
(ii) ( X  Y ) n   Crn X n r Y r
for
n = 3, 4, 5, ... .
r0
(Note: For any square matrix A , define A 0  I .)
(3 marks)
 1 2 4
(b) By using (a)(ii) and considering  0 1 3 , or otherwise, find


 0 0 1
 1 2 4
 0 1 3


 0 0 1
100
.
(4 marks)
(c) If X and Y are square matrices,
(i) prove that ( X  Y ) 2  X 2  2 XY  Y 2 implies XY = YX ;
(ii) prove that ( X  Y ) 3  X 3  3 X 2 Y  3 XY 2  Y 3 does NOT
implies XY = YX .
(Hint : Consider a particular X and Y, e.g. X  
1 0
 b 0 .)
, Y 

 1 0
 0 0
[HKAL90] (8 marks)
Prepared by K. F. Ngai
Page 14
Matrices and Determinants
Advanced Level Pure Mathematics
Prepared by K. F. Ngai
Page 15
Matrices and Determinants
Advanced Level Pure Mathematics
8.4
INVERSE OF A SQUARE MATRIX
N.B.
(1) If a, b, c are real numbers such that ab=c and b is non-zero, then
c
1
 cb 1 and b is usually called the multiplicative inverse of b.
b
C
(2) If B, C are matrices, then
is undefined.
B
a
Def.
A square matrix A of order n is said to be non-singular or invertible if and only
if there exists a square matrix B such that AB = BA = I.
The matrix B is called the multiplicative inverse of A, denoted by A 1
i.e.
AA1  A1 A  I .
3 5
 2 5 .
 , show that the inverse of A is 

 1 2
 1 3 
e.g.12 Let A  
i.e.
 3 5


 1 2
1
 2 5 .


 1 3 
2 5


 1 3
e.g.13 Is 
1
 3 5 ?


 1 2
Prepared by K. F. Ngai
Page 16
Matrices and Determinants
Advanced Level Pure Mathematics
Def.
If a square matrix A has an inverse, A is said to be non-singular or invertible.
Otherwise, it is called singular or non-invertible.
e.g.
 3 5 and  2 5 are both non-singular.




 1 2
 1 3 
i.e.
A is non-singular iff A 1 exists.
Thm.
The inverse of a non-singular matrix is unique.
N.B.
(1) I 1  I , so I is always non-singular.
(2) OA = O  I , so O is always singular.
(3) Since AB = I implies BA = I.
Hence proof of either AB = I or BA = I is enough to assert that B is the
inverse of A.
e.g.14 Let A  
2 1
.
 7 4
(a) Show that I  6 A  A 2  O .
(b) Show that A is non-singular and find the inverse of A.
(c) Find a matrix X such that AX  
1 1
.
 1 0
Prepared by K. F. Ngai
Page 17
Matrices and Determinants
Advanced Level Pure Mathematics
Properties of Inverses
Thm.
Let A, B be two non-singular matrices of the same order and  be a scalar.
(a) ( A1 )1  A .
(b) AT is a non-singular and ( AT )1  ( A1 )T .
(c) A n is a non-singular and ( A n ) 1  ( A 1 ) n .
(d) A is a non-singular and (A) 1 
1

A 1 .
(e) AB is a non-singular and ( AB) 1  B 1 A 1 .
Proof
Refer to Textbook P.228.
Prepared by K. F. Ngai
Page 18
Matrices and Determinants
Advanced Level Pure Mathematics
8.5
DETERMINANTS
Def.
Let A  aij
 
be a square matrix of order n. The determinant of A, detA or |A|
is defined as follows:
a
a12
(a) If n=2, det A  11
 a11 a22  a12 a21
a21 a22
a11 a12 a13
(b) If n=3, det A  a21 a22 a23
a 31 a32 a33
or det A  a11 a22 a33  a21 a32 a13  a31 a12 a23
 a31 a22 a13  a32 a23 a11  a33 a21 a12
e.g.15 Evaluate
1 3
(a)
4 1
3
1 2
(b) det  2 1 0 
1 2 1


3 2 x
e.g.16 If 8 x 1  0 , find the value(s) of x.
3 2 0
Prepared by K. F. Ngai
Page 19
Matrices and Determinants
Advanced Level Pure Mathematics
N.B.
a11
det A  a 21
a 31
 a11
or
a12
a22
a32
a22
a32
  a12
a21
a31
a13
a23
a33
a23
a
 a12 21
a33
a31
a23
a
 a22 11
a33
a31
a23
a
 a13 21
a33
a31
a13
a
 a32 11
a33
a21
a22
a32
a13
a23
or . . . . . . . . .
  
By using   
  
e.g.17 Evaluate
3 2 0
(a) 0 1 1
0 2 3
0 2 0
(b) 8 2 1
3 2 3
Prepared by K. F. Ngai
Page 20
Matrices and Determinants
Advanced Level Pure Mathematics
8.6
PROPERTIES OF DETERMINANTS
a1
(1) a2
a3
b1
b2
b3
c1 a1
c2  b1
c3 c1
a2
b2
c2
a3
b3
c3
i.e. det( A T )  det A .
a1 b1
(2) a2 b2
a3 b3
c1
b1
c2   b2
c3
b3
a1
a2
a3
c1 b1
c2  b2
c3 b3
c1
c2
c3
a1
a2
a3
a1
a2
a3
c1
a2
c2   a1
c3
a3
b2
b1
b3
c2 a 2
c1  a3
c3 a1
b2
b3
b1
c2
c3
c1
b1
b2
b3
a1
(3) a2
a3
0 c1
a1
0 c2  0  a 2
0 c3
0
a1
(4) a2
a3
a1
a2
a3
c1
a1
c2  0  a1
c3
a3
b1
b2
0
c1
c2
0
b1
b1
b3
c1
c1
c3
a1
a1 a2 a3
(5) If

 , then a2
b1 b2 b3
a3
b1
b2
b3
a1  x1
(6) a2  x 2
a3  x 3
c1 x1
c2  x 2
c3 x 3
pa1
(7) pa2
pa3
pa1
pa2
pa3
N.B.
b1
b2
b3
b1
b2
b3
pb1
pb2
pb3
c1 a1
c2  a 2
c3 a 3
c1
a1
c2  p a 2
c3
a3
b1
b2
b3
b1
b2
b3
pc1
a1
3
pc2  p a2
pc3
a3
 pa1
(1)  pa2

 pa3
pb1
pb2
pb3
c1
c2  0
c3
c1
a1
c2  pa2
c3
a3
b1
b2
b3
b1
b2
b3
c1
c2
c3
b1
pb2
b3
c1
pc2
c3
c1
c2
c3
pc1 
 a1

pc2  p a2


pc3 
 a3
b1
b2
b3
c1 
c2 

c3 
(2) If the order of A is n, then det(A)  n det( A)
Prepared by K. F. Ngai
Page 21
Matrices and Determinants
Advanced Level Pure Mathematics
a1
(8) a2
a3
b1
b2
b3
c1
a1  b1
c2  a2  b2
c3 a3  b3
N.B.
x1
x2
x3
y1
y2
y3
e.g.18 Evaluate
b1
b2
b3
c1
c2
c3
z1
x1  y1  z1
C2  C3  C1
z2
x   y 2  z 2
 2
z3
x 3  y3  z3
1 2 0
(a) 0 4 5 ,
6 7 8
y1
y2
y3
z1
z2
z3
5 3 7
(b) 3 7 5
7 2 6
1 a b c
e.g.19 Evaluate 1 b c  a
1 c ab
Prepared by K. F. Ngai
Page 22
Matrices and Determinants
Advanced Level Pure Mathematics
e.g.20 Factorize the determinant
x
y
xy
y
xy
x
xy
x
y
e.g.21 Factorize each of the following :
a 3 b3 c 3
(a) a b c
1 1 1
[HKAL91] (4 marks)
2a 3
2b 3
2c 3
(b) a 2
b2
c2
1  a 3 1  b3 1  c 3
Prepared by K. F. Ngai
Page 23
Matrices and Determinants
Advanced Level Pure Mathematics
Def.
Multiplication of Determinants.
a
a12
b b
Let A  11
, B  11 12
a 21 a 22
b21 b22
a11 a12 b11 b12
a 21 a 22 b21 b22
a b  a12 b21 a11b12  a12 b22
 11 11
a 21b11  a 22 b21 a 21b12  a 22 b22
Then A B 
Properties :
(1) det(AB)=(detA)(detB)
i.e. AB  A B
(2) |A|(|B||C|)=(|A||B|)|C|
N.B.
A(BC)=(AB)C
(3) |A||B|=|B||A|
N.B.
ABBA in general
(4) |A|(|B|+|C|)=|A||B|+|A||C|
N.B.
A(B+C)=AB+AC
Prepared by K. F. Ngai
Page 24
Matrices and Determinants
Advanced Level Pure Mathematics
1
e.g.22 Prove that a
a2
1
b
b2
1
c  (a  b)(b  c)(c  a )
c2
Prepared by K. F. Ngai
Page 25
Matrices and Determinants
Advanced Level Pure Mathematics
Minors and Cofactors
 a11
Let A   a 21

 a 31
Def.
A11 
Since
a 22
a 32
a 23
a 33
A  a 21
a13 
a 23  , then Aij , the cofactor of aij , is defined by

a 33 
a
a
a
a
, A12   21 23 , ... , A33  11 12 .
a 31 a 33
a 21 a 22
a12
a32
a12
a 22
a 32
a13
a33
+ a 22
a11
a 31
a13
a
a 23 11
a 33
a 31
a12
a 32
 a21 A21  a22 A22  a23 A23
Thm.
(a)
det A if i  j
a i1 A j1  a i 2 A j 2  a i 3 A j 3  
if i  j
0
(b)
det A if i  j
a1i A1 j  a2i A2 j  a3i A3 j  
if i  j
0
e.g. a11 A11  a12 A12  a13 A13  det A , a11 A21  a12 A22  a13 A23  0 , etc.
 a11
a12

 a 31
a 32
e.g.23 Let A   a 21 a 22
a13 
a 23  and cij be the cofactor of aij , where 1  i , j  3 .

a 33 
 c11
(a) Prove that A c12

 c13
c21
c22
c23
c11
(b) Hence, deduce that c12
c13
c31 
c32   (det A) I

c33 
c21
c22
c23
c31
c32  (det A) 2
c33
Prepared by K. F. Ngai
Page 26
Matrices and Determinants
Advanced Level Pure Mathematics
8.7
INVERSE OF SQUARE MATRIX BY DETERMINANTS
Def.
 A11
The cofactor matrix of A is defined as cofA   A21

 A31
Def.
The adjoint matrix of A is defined as
 A11
adjA  ( cofA)   A12

 A13
T
A21
A22
A23
A12
A22
A32
A13 
A23  .

A33 
A31 
A32  .

A33 
e.g.24 If A  
a b
 , find adjA.
 c d
e.g.25
 1 1 3
(a) Let A   1 2 0  , find adjA.


1 1 1 
3 2 1 
(b) Let B   1 1 1 , find adjB.


 5 1 1
Prepared by K. F. Ngai
Page 27
Matrices and Determinants
Advanced Level Pure Mathematics
Thm.
For any square matrix A of order n ,
A(adjA) = (adjA)A = (detA)I
 a11 a12  a1n   A11
 a a  a  A
2n
A( adjA)   21 22
  12

  
 
 an1 an2  ann   A1n
A21  An1 
A22  An2 


 
A2 n  Ann 
Prepared by K. F. Ngai
Page 28
Matrices and Determinants
Advanced Level Pure Mathematics
Thm.
Let A be a square matrix. If detA  0 , then A is non-singular
and
A1 
1
 adjA .
det A
Proof
Let the order of A be n , from the above theorem ,
e.g.26
3 2 1 
Given that A   1 1 1 , find A 1 .


 5 1 1
1
 AadjA  I
det A
e.g.27 Suppose that the matrix A  
a b
1
 is non-singular , find A .
 c d
Prepared by K. F. Ngai
Page 29
Matrices and Determinants
Advanced Level Pure Mathematics
e.g.28 Given that A  
3 5
,
 1 2
Thm.
find A 1 .
A square matrix A is non-singular iff detA  0 .
e.g.29 Show that A  
3 5
 is non-singular.
 1 2
Prepared by K. F. Ngai
Page 30
Matrices and Determinants
Advanced Level Pure Mathematics
e.g.30
 x  1 2 x  1
Let A   x  1 2 1  , where x  R .


7 x 
 5
(a) Find the value(s) of x such that A is non-singular.
(b) If x=3 , find A 1 .
N.B.
A is singular (non-invertible) iff A 1 does not exist.
Thm.
A square matrix A is singular iff detA = 0.
Prepared by K. F. Ngai
Page 31
Matrices and Determinants
Advanced Level Pure Mathematics
Properties of Inverse matrix.
Let A, B be two non-singular matrices of the same order and  be a scalar.
(1) (A) 1 
1

A 1
(2) ( A 1 ) 1  A
(3) ( A T ) 1   A 1 
(4) ( A n ) 1   A 1 
T
n
for any positive integer n.
(5) ( AB ) 1  B 1 A 1
(6) The inverse of a matrix is unique.
(7) det( A 1 ) 
N.B.
1
det A
XY  0 
 X  0 or Y  0
(8) If A is non-singular , then AX  0  A1 AX  A0  0
X 0
N.B.
XY  XZ 
 X  0 or Y  Z
(9) If A is non-singular , then AX  AY  A1 AX  A1 AY
X Y
(10) ( A 1 MA) n  ( A 1 MA)( A 1 MA) ( A 1 MA)
 a 1
 a 0 0
(11) If M   0 b 0 , then M 1   0



 0 0 c
 0
0
b 1
0
 A 1 M n A
0
0.

c 1 
 an 0 0 
 a 0 0
(12) If M   0 b 0 , then M n   0 b n 0  where n  0 .



n
 0 0 c
0 0 c
 4 1 0
 1  3 1 
 1 0 0

e.g.31 Let A   1 3 1 , B   0 13
and
4
M   0 1 0 .






 0 3 1
 0 33 10
 0 0 2
(a) Find A 1 and M 5 .
(b) Show that ABA 1  M .
(c) Hence, evaluate B 5 .
Prepared by K. F. Ngai
Page 32
Matrices and Determinants
Advanced Level Pure Mathematics
3 8
 2 4 .

 and P  
1 1 
 1 5
e.g.32 Let A  
(a) Find P 1 AP .
(b) Find A n , where n is a positive integer.
[HKAL94] (6 marks)
Prepared by K. F. Ngai
Page 33
Matrices and Determinants
Advanced Level Pure Mathematics
e.g.33 (a) Show that if A is a 3x3 matrix such that A   A , then detA=0.
t
 1 2 74 
(b) Given that B   2 1 67 ,


 74 67 1 
use (a) , or otherwise , to show det(I  B)  0 .
Hence deduce that det( I  B 4 )  0 .
[HKAL93] (7 marks)
e.g.34 (a) If  ,  and  are the roots of x 3  px  q  0 , find a cubic equation whose
roots are  2 ,  2 and  2 .
x 2 3
(b) Solve the equation 2 x 3  0 .
2 3 x
Hence, or otherwise, solve the equation
x 3  38x 2  361x  900  0 .
[HKAL94] (6 marks)
Prepared by K. F. Ngai
Page 34
Matrices and Determinants
Advanced Level Pure Mathematics
e.g.35 Let M be the set of all 2x2 matrices. For any A  
a11
 a21
a12 
 M ,
a22 
define tr ( A)  a11  a22 .
(a) Show that for any A, B, C  M and ,   R,
(i) tr (A  B )  tr ( A)  tr ( B ) ,
(ii) tr ( AB)  tr (BA) ,
(iii) the equality “ tr ( ABC)  tr (BAC) ” is not necessary true.
(5 marks)
(b) Let A  M.
(i) Show that A 2  tr ( A) A  (det A) I ,
where I is the 2x2 identity matrix.
(ii) If tr ( A 2 )  0 and tr ( A)  0 , use (a) and (b)(i) to show that
A is singular and A 2  0 .
(5 marks)
(c) Let S, T  M such that (ST  TS )S  S (ST  TS ) .
Using (a) and (b) or otherwise, show that
( ST  TS ) 2  0
[HKAL92] (5 marks)
Prepared by K. F. Ngai
Page 35
Matrices and Determinants
Advanced Level Pure Mathematics
Prepared by K. F. Ngai
Page 36
Matrices and Determinants
Advanced Level Pure Mathematics
e.g.36 Eigenvalue and Eigenvector
3 1
 and let x denote a 2x1 matrix.
2 0 
Let A  
(a) Find the two real values  1 and  2 of  with  1 >  2
such that the matrix equation
(*)
Ax  x
has non-zero solutions.
(b) Let x1 and x2 be non-zero solutions of (*) corresponding to
 1 and  2 respectively. Show that if
x 
x1   11 
 x21 
x 
x2   12 
 x22 
and
then the matrix X  
x11
 x21
x12 
 is non-singular.
x22 
(c) Using (a) and (b), show that
and hence
  1n
A  X
 0
n
0

AX  X  1

 0 2
0  1 where n is a positive integer.
X
 2n 
3 1
Evaluate 
 .
2 0 
n
[HKAL82]
Prepared by K. F. Ngai
Page 37