Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
Mathematics4
Matrices and Determinants
INTRODUCTION : MATRIX / MATRICES
2
SOME SPECIAL MATRIX
3
ARITHMETRICS OF MATRICES
4
INVERSE OF A SQUARE MATRIX
16
DETERMINANTS
19
PROPERTIES OF DETERMINANTS
21
INVERSE OF SQUARE MATRIX BY DETERMINANTS
27
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Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
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
a2  a n  is called a row matrix or row vector.
2
e.g.  7  is a column vector of order 31.
3
 
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e.g.  2 3 4 is a row vector of order 13.
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 , a22 , , ann is called the principal diagonal.
3 9
e.g. 
is a square matrix of order 2.
0 2
 
7. Notation : aij
m n
a 
,
ij
m n
, A , ...
SOME SPECIAL MATRIX.
Def.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..2 Let A  aij
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


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2 3
e.g. 
is an upper triangular matrix.
 0 5 
 
Def..3 Let A  aij
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..4 If A is a diagonal matrix and a11  a22  ann  1 , then A is called an identity
matrix or a unit matrix, denoted by I n .
1 0 ,
I2  
0 1
e.g.
1 0 0
I 3  0 1 0
0 0 1


ARITHMETRICS OF MATRICES.
Def..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
e.g.
a 2 1 c
 4 b   d 1
N.B.
2 3 2 4
4 0   3 0
 
Def..6 Let A  aij
m n
m n
 aij  bij for all i , j .

and
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 
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2
3
1

2
4
1
2 3 1 is not defined.
N.B.
1.
0  
1 0 4 
4 
3
2.
 5 is not defined.
0
Def..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 . Find -A and A-B.
If A  
and
B

3 1 1
1 0 2
Thm..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
Scalar Multiplication.
Let A  aij mn , k is scalar. Then kA is the matrix C  cij
 
cij  kaij ,
i, j .
 
i.e. kA  kaij
e.g.
m n
defined by
m n
 3 2
If A  
,
5 6 
then
N.B.
 
-2A=
;
3
A
2
(1) -A=(-1)A
(2) A-B=A+(-1)B
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Thm..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)
Def..9
Let A  aij mn . The transpose of A, denoted by A T , or A  , is
 
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.
Thm..3
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 
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..11 A square matrix A is called a symmetric matrix iff A T  A .
i.e.
e.g.
e.g.
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.
Def..12 A square matrix A is called a skew-symmetric matrix iff A T   A .
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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 aii  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
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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 0 . Find AB. Is BA well defined?
2 1 2 2
1 4

 3 2
N.B.
Thm..4
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)
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(g) ( AB) T  B T A T .
N.B.
(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  2 B )C
(b) ( AC ) 2
(c) ( B T  3C ) T D
(d) (2A) T B  DD T
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2nd Class – Mechanical Eng. Dep.
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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  
 such that
  
 2 1
 2 1 .
A T  A and 
 A  A

 3 0
 3 0
3 1  1   0  1
(c) If 
   
   , find the values of x and  .
 1 1   x  0    x
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e.g.8
e.g.9
cos  sin  
Let A  
 . Prove by mathematical induction that
 sin  cos 
 cos n  sin n 
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   a

0
n
a n  bn 

a  b  for all positive integers n.
n
b

95
(b) Hence, or otherwise, evaluate  1 2 .
 0 3
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 0 1 0
e.g.10 (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.
A2  O ),
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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 .)
 1 2 4
(b) By using (a)(ii) and considering  0 1 3 , or otherwise, find


 0 0 1
100
 1 2 4
 0 1 3 .


 0 0 1
(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
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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
 cb 1 and b 1 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 .
5
.
 1 3 
e.g.12 Let A   3 5 , show that the inverse of A is  2
 1 2
1
 2 5 .


 1 3 
i.e.
 3 5


 1 2
e.g.13
Is  2 5
 1 3 
1
 3 5 ?


 1 2
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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 if 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  6A  A2  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 Eng. Baseem Adnan Al-twajre
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Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
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) An is a non-singular and ( An ) 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 .
Prepared by Eng. Baseem Adnan Al-twajre
Page 18
Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
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
a31 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
1 2 3 
(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
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Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
N.B.
a11
det A  a21
a31
 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
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Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
PROPERTIES OF DETERMINANTS
a1
(1) a2
a3
b1
b2
b3
c1 a1
c2  b1
c3 c1
a2
b2
c2
a1
(2) a2
a3
b1
b2
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
b1
b2
b3
c1
a2
c2   a1
c3
a3
b2
b1
b3
c2 a 2
c1  a3
c3 a1
b2
b3
b1
c2
c3
c1
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
a3
b3
c3
b1
b2
0
b1
b1
b3
i.e. det( AT )  det A .
c1
c2
0
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)
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Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
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  y2  z2
 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
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Matrices and Determinants
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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
2a 3
2b 3
2c 3
(b) a 2
b2
c2
1  a 3 1  b3 1  c 3
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Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
Def.
Multiplication of Determinants.
a
a12
b b
Let A  11
, B  11 12
a21 a22
b21 b22
a11 a12 b11 b12
a21 a22 b21 b22
a b a b
a11b12  a12 b22
 11 11 12 21
a21b11  a22 b21 a21b12  a22 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
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Page 24
Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
1
e.g.22 Prove that a
a2
1
b
b2
1
c  (a  b)(b  c)(c  a )
c2
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Page 25
Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
Minors and Cofactors
 a11
a13 
a 23  , then Aij , the cofactor of a ij , is defined by


 a 31 a 32 a 33 
a 23
, A12   a 21 a 23 , ... , A33  a11 a12 .
a 31 a 33
a 33
a 21 a 22
a12
Let A   a 21 a 22
Def.
A11 
Since
a 22
a 32
A  a 21
a12
a13
a32
a33
+ a 22 a11
a 31
a13
a
a 23 11
a 33
a 31
a12
a 32
 a21A21  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.
e.g.23
 a11
Let A   a 21

 a 31
a12
a 22
a 32
a13 
a 23  and cij be the cofactor of a ij , 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 Eng. Baseem Adnan Al-twajre
Page 26
Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
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
1 1 3
0  , find adjA.


1 1 1 
e.g.25 (a) Let A  1 2
3 2
1
(b) Let B   1 1 1 , find adjB.


 5 1 1
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Page 27
Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
Thm.
For any square matrix A of order n ,
A(adjA) = (adjA)A = (detA)I
 a11
a
A( adjA)   21
 
 an1
a12  a1n   A11
a22  a2 n   A12


  
an2  ann   A1n
A21  An1 
A22  An2 


 
A2 n  Ann 
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Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
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
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Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
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
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Matrices and Determinants
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 x  1 2 x  1
1  , where x  R .


7 x 
 5
e.g.30 Let A   x  1 2
(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) if A 1 does not exist.
Thm.
A square matrix A is singular iff detA = 0.
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Matrices and Determinants
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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)
(11)
 a 1
 a 0 0
If M   0 b 0 , then M 1   0



 0 0 c
 0
(12)
e.g.31
0
b 1
0
 A 1 M n A
0
0.

c 1 
 an 0 0 
 a 0 0
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




Let A  1 3 1 , B  0 13
and M   0 1 0 .
4






 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 .
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Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
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.
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Matrices and Determinants
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e.g.33 (a) Show that if A is a 3x3 matrix such that A t   A , then detA=0.
2 74 
1 67 ,


 74 67 1 
 1
(b) Given that B   2
use (a) , or otherwise , to show det(I  B)  0 .
Hence deduce that det( I  B 4 )  0 .
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 .
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Matrices and Determinants
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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.
(b) Let A  M.
(i) Show that A2  tr ( A) A  (det A) I ,
where I is the 2x2 identity matrix.
(ii) If tr ( A2 )  0 and tr ( A)  0 , use (a) and (b)(i) to show that
A is singular and A2  0 .
(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
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Matrices and Determinants
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Matrices and Determinants
2nd Class – Mechanical Eng. Dep.
e.g.36 Eigenvalue and Eigenvector
Let A   3 1 and let x denote a 2x1 matrix.
2
0
(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 
and
x 
x2   12 
 x22 
then the matrix X   x11 x12  is non-singular.
 x21
x22 
(c) Using (a) and (b), show that
  1n
A  X
 0
and hence
n
0

AX  X  1

 0 2
0  1
 X where n is a positive integer.
 2n 
Evaluate  3 1 .
n
2
0
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