Matrices

DEFINITION
A rectangular arrangement of elements in rows and columns, is called a matrix. Such a rectangular arrangement
of numbers is enclosed by small ( ) or big [ ] brackets. Generally a matrix is represented by a capital latter A, B,
C......... etc. and its element are represented by small letters a, b, c, x, y etc.
Following are some examples of a matrix :
a b  1 5 3  2
A =  c d  , B = 4 0 2 C =  0  , D = [1, 5, 6], E = [5]
     

ORDER OF MATRIX
A matrix which has m rows and n columns is called a matrix of order m × n, and its represented by
Am×n or A = [aij]m×n
It is obvious to note that a matrix of order m × n contains mn elements. Every row of such a matrix contains n
elements and every column contains m elements.

TYPES OF MATRICES
Row matrix
If in a matrix, there is only one row, then it is called a Row Matrix.
Thus A = [aij]m×n is a row matrix if m = 1
Column Matrix
If in a matrix, there is only one column, then it is called a column matrix.
Thus A = [aij]m×n is a column matrix if n = 1.
Square matrix
If number of rows and number of columns in a matrix are equal, then it is called a square matrix.
Thus A = [aij]m×n is a square matrix if m = n.
Note : (a) If m  n then matrix is called a rectangular matrix.
(b) The elements of a square matrix A for which i = j i.e., a11, a22, a33,......ann are called principal diagonal
elements and the line joining these elements is called the principal diagonal or leading diagonal of
matrix A.
(c) Trace of a matrix : The sum of principal diagonal elements of a square matrix A is called the trace of
matrix A which is denoted by trace A. Trace A = a11  a 22  ....a nn
Singleton matrix
If in a matrix there is only one element then it is called singleton matrix.
Thus A = [aij]m×n is a singleton matrix if m = n = 1.
Null or zero matrix
If in a matrix all the elements are zero then it is called a zero matrix and it is generally denoted by O.
Thus A = [aij]m×n is a zero matrix if aij = 0 for all i and j.
Diagonal matrix
If all elements except the principal diagonal in a square matrix are zero, it is called a diagonal matrix.
Thus a square matrix A = [aij] is a diagonal matrix if aij = 0, when i  j.

[1]
[2] Matrices and Determinants

Note : (a) No element of principal diagonal in diagonal matrix is zero.

(b) Number of zero is a diagonal matrix is given by n2 – n where n is a order of the matrix.
Scalar Matrix
If all the elements of the diagonal in a diagonal matrix are equal, it is called a scalar matrix.
Thus a square matrix A [aij] is a scalar matrix is
0 i  j
aij = k i  j where k is a constant.

Unit matrix
If all elements of principal diagonal in a diagonal matrix are 1, then it is called unit matrix. A unit matrix of order
n is denoted by In.
Thus a square matrix
1 i  j
A = [aij] is a unit matrix if aij = 0 i  j

Note : Every unit matrix is a scalar matrix.
Triangular matrix
A square matrix [aij] is said to be triangular if each element above or below the principal diagonal is zero. It is of
two types -
(a) Upper triangular matrix : A square matrix [aij] is called the upper triangular matrix, if aij = 0 when i > j.
(b) Lower triangular matrix : A square matrix [aij] is called the lower triangular matrix, if
aij = 0 when i < j
n(n  1)
Note : Minimum number of zero in a triangular matrix is given by where n is order of matrix.
2

Equal matrix
Two matrices A and B are said to be equal if they are of same order and their corresponding elements are equal.
Singular matrix
Matrix A is said to be singular matrix if its determinant |A| = 0, otherwise non-singular matrix i.e.,
If det |A| = 0  singular and det |A|  0  non-singular

ADDITION AND SUBTRACTION OF MATRICES

If A = [aij]m×n and B = [bij]m×n are two matrices of the same order then their sum A + B is a matrix whose each
element is the sum of corresponding elements.
i.e., A + B = [aij + bij ] m×n
A – B is defined as A – B = [aij – bij]m×n
Note : Matrix addition and subtraction can be possible only when matrices are of same order.
Properties of matrices addition
If A, B and C are matrices of same order, then-
(i) A + B = B + A (Commutative Law)
(ii) (A + B) + C = A + (B + C) (Associative law)
(iii) A + O = O + A = A, where O is zero matrix which is additive identity of the matrix.
(iv) A + (–A) = 0 = (–A) + A where (–A) is obtained by changing the sign of every element of A which is
additive inverse of the matrix
Matrices and Determinants [3]

A  B  A  C
(v) B  A  C  A   B = C (Cancellation law)

(vi) Trace (A ± B) = trace (A) ± trace (B)

SCALAR MULTIPLICATION OF MATRICES

Let A = [aij]m×n be a matrix and k be a number then the matrix which is obtained by multiplying every element of
A by k is called scalar multiplication of A by k and it denoted by kA.
Thus A = [aij]m×n  kA = [kaij]m×n

Properties of scalar multiplication

If A, B are matrices of the same order and m, n are any numbers, then the following results can be easily
established.
(i) m(A + B) = mA + mB (ii) (m + n)A = mA + nA (iii) m(nA) = (mn)A = n(mA)

MULTIPLICATION OF MATRICES
If A and B be any two matrices, then their product AB will be defined only when number of column in A is equal
to the number of rows in B. If A = [aij]m×n and B = [bij]n×p then their product AB = C = [cij], will be matrix of order
m × p, where
n

(AB)ij = Cij  a b
r 1
ir rj

Properties of matrix multiplication

If A, B and C are three matrices such that their product is defined, then
(i) AB  BA (Generally not commutative)
(ii) (AB) C = A (BC) (Associative Law)
(iii) IA = A = AI I is identity matrix for matrix multiplication
(iv) A (B + C) = AB + AC (Distributive law)
(v) If AB = AC   B = C (cancellation Law is not applicable)
(vi) If AB = 0 It does not mean that A = 0 or B = 0, again product of two non-zero matrix may be zero
matrix.
(vii) trance (AB) = trance (BA)
Note : (i) The multiplication of two diagonal matrices is again a diagonal matrix.
(ii) The multiplication of two triangular matrices is again a triangular matrix.
(iii) The multiplication of two scalar matrices is also a scalar matrix.
(iv) If A and B are two matrices of the same order, then
(a) (A + B)2 = A2 + B2 + AB + BA
(b) (A – B)2 = A2 + B2 – AB – BA
(c) (A – B) (A + B) = A2 – B2 + AB – BA
(d) (A + B) (A – B) = A2 – B2 – AB + BA
(e) A (–B) = (–A) B = –(AB)
Positive Integral powers of a matrix
The positive integral powers of a matrix A are defined only when A is a square matrix.
Also then A2 = A.A A3 = A.A.A = A2A
[4] Matrices and Determinants

Also for any positive integers m, n

(i) Am An = Am+n (ii) (Am)n = Amn = (An)m
(iii) In = I, Im = I (iv) A° = In where A is a square matrices of order n.

TRANSPOSE OF MATRIX
If we interchange the rows to columns and columns to rows of a matrix A, then the matrix so obtained is called
the transpose of A and it is denoted by
AT or At or A'
From this definition it is obvious to note that
(i) Order of A is m × n  order of AT is n × m
(ii) (AT)ij = (A)ji, " i, j)
Properties of Transpose
If A, B are matrices of suitable order then
(i) (AT)T = A
(ii) (A + B)T = AT + BT
(iii) (A – B)T = AT – BT
(iv) (kA)T = kAT
(v) (AB)T = BTAT
(vi) (A1A2.....An)T = AnT.....A2TA1T
(vii) (An)T = (AT)n, n  N

SYMMETRIC AND SKEW-SYMMETRIC MATRIX

(a) Symmetric matrix : A square matrix A = [aij] is called symmetric matrix if aij = aji for all ˆi  ˆj or AT = A.
Note : (i) Every unit matrix and square zero matrix are symmetric matrices.
n(n  1)
(ii) Maximum number of different element in a symmetric matrix is
2
(b) Skew-symmetric matrix : A square matrix A = [aij] is called skew-symmetric matrix if
aij = – aji for all i, j or AT = –A
Note : (i) All principal diagonal elements of a skew-symmetric matrix are always zero because for
any diagonal element - aii = – aii  aii = 0
(ii) Trace of a skew symmetric matrix is always 0
Properties of symmetric and skew-symmetric matrices
(i) If A is a square matrix, then A + AT, AAT, ATA are symmetric matrices while A–AT is skew-symmetric
matrices.
(ii) If A, B are two symmetric matrices, then-
(a) A ± B, AB + BA are also symmetric matrices.
(b) AB – BA is a skew-symmetric matrix.
(c) AB is a symmetric matrix when AB = BA
(iii) If A, B are two skew-symmetric matrices, then-
(a) A ± B, AB – BA are skew-symmetric matrices.
(b) AB + BA is a symmetric matrix.
Matrices and Determinants [5]

(iv) If A is a skew-symmetric matrix and C is a column matrix, then CT AC is a zero matrix.

(v) Every square matrix A can be uniquely be expressed as sum of a symmetric and skew symmetric matrix i.e.,

1 T  1 T 
A =  2 ( A  A )   2 ( A  A )
   

DETERMINANT OF A MATRIX

a11 a12 a13 

 
If A = a 21 a 22 a 23 
be a square matrix, then its determinant, denoted by |A| or det. (A) is
a 31 a 32 a 33 

a11 a12 a13 

 
defined as |A| = a 21 a 22 a 23 
a 31 a 32 a33 

Properties of the determinant of a matrix

(i) |A| exist  A is a square matrix
(ii) |AB| = |A| |B|
(iii) |AT| = |A|
(iv) |kA| = kn |A|, if A is a square matrix of order n.
(v) If A and B are square matrices of same order then |AB| = |BA|
(vi) If A is skew symmetric matrix of odd order then |A| = 0
(vii) If A = diag (a1, a2, .....an) then |A| = a1a2....an
(viii) |A|n = |An|, n  N

ADJOINT OF A MATRIX
If every element of a square matrix A be replaced by its cofactor in |A|, then the transpose of the matrix so
obtained is called the adjoint of A and it is denoted by adj A
Thus if A = [aij] be a square matrix and Cij be the cofactor of aij in |A|, then
adj A = [Cij]T
 (adj A)ij = Cij
T
 a 11 a 12 .... a 1n   C11 C12 .... C1n   C11 C 21 .... C n1 
a C  C .... Cn 2 
 21 a 22 .... a 2 n   21
C 22 .... C 2n
   12
C 22

Hence if A =  .... .... .... ....  , then adj A =  .... .... .... ....   .... .... .... .... 
     
 a n1 a n2 .... a nn 
 C n1 Cn 2 .... C nn  C1n C 2n .... Cnn 

Properties of Adjoint Matrix

If A, B are square matrices of order n and In is corresponding unit matrix, then
(i) A (adj A) = |A| In = (adj A) A
(Thus A (adj A) is always a scalar matrix)
(ii) |adj A| = |A|n–1
(iii) adj (adj A) = |A|n–2 A
[6] Matrices and Determinants

2
(iv) |adj (adj A)| = | A |(n1) (v) adj (AT) = (adj A)T
(vi) adj (AB) = (adj B) (adj A) (vii) adj (Am) = (adj A)m, m  N
(viii) adj (kA) = kn–1 (adj A), k  R (ix) adj (In) = In
(x) adj 0 = 0 (xi) A is symmetric  adj A is also symmetric.
(xii) A is diagonal  adj A is also diagonal. (xiii) A is triangular  adj A is also triangular..
(xiv) A is singular  |adj A| = 0

INVERSE MATRIX
If A and B are two matrices such that
AB = I = BA
then B is called the inverse of A and it is denoted by A–1. Thus
A–1 = B  AB = I = BA
Further we may note from above property (i) of adjoint matrix that if |A|  0, then
adj ( A ) (adj A ) 1
A  A  A–1 = | A | adj A
|A| |A|
Thus A–1 exists  |A|  0.
Note :
(i) Matrix A is called invertible if A–1 exists.
(ii) Inverse of a matrix is unique.
Properties of Inverse Matrix
(i) (A–1)–1 = A
(ii) (AT)–1 = (A–1)T
(iii) (AB)–1 = B–1A–1
(iv) (An)–1 = (A–1)n, n  N
(v) adj (A–1) = (adj A)–1
1
(vi) |A–1| = | A | = |A|–1

(vii) A = diag (a1,a2, ...., an)  A–1 = diag (a1–1 , a2–1, ....., an–1)
(viii) A is symmetric  A–1 is also symmetric.
(ix) A is diagonal |A|  0  A–1 is also diagonal.
(x) A is scalar matrix  A–1 is also scalar matrix.
(xi) A is triangular |A|  0  A–1 is also triangular..

SOME IMPORTANT CASES OF MATRICES

Orthogonal Matrix
A square matrix A is called orthogonal if
AAT = I = ATA ; i.e., if A–1 = AT
Idempotent matrix
A square matrix A is called an idempotent matrix if A2 = A
Matrices and Determinants [7]

Involutory Matrix
A square matrix A is called an involutory matrix if A2 = I or A–1 = A
Nilpotent matrix
A square matrix A is called a nilpotent matrix if there exist a p  N such that AP = 0

Hermition matrix

A square matrix A is skew-Hermition matrix if Aq = A ; i.e., aij = – aji " i, j

Skew hermitian matrix

A square matrix A is skew-hermition is A = –Aq i.e., aij = – aji " i, j

Period of a matrix
If for any matrix A Ak+1 = A
then k is called period of matrix (where k is a least positive integer)
Differentiation of matrix
 f ( x ) g( x )
If A = h( x ) l( x )
 
dA  f ' ( x ) g' ( x )
then dx  h' ( x ) l' ( x ) is a differentiation of matrix A
 
Submatrix
Let A be m × n matrix, then a matrix obtained by leaving some rows or columns or both of a is called a sub matrix
of A
Rank of a matrix
A number r is said to be the rank of a m × n matrix A if
(a) every square sub matrix of order (r + 1) or more is singular and
(b) there exists at least one square submatrix of order r which is non-singular.
Thus, the rank of matrix is the order of the highest order non-singular sub matrix.
5 6 
We have |A| = 0 therefore r (A) is less then 3, we observe that 4 5  is a non-singular square sub matrix of
 
order 2 hence r (A) 2.
Note :
(i) The rank of the null matrix is zero.
(ii) The rank of matrix is same as the rank of its transpose i.e., r(A) = r(AT)
(iii) Elementary transformation of not alter the rank of matrix.
[8] Matrices and Determinants

DETERMINANTS

DEFINITION
When an algebraic or numerical expression is expressed in a square form containing some rows and columns, this
square form is named as a determinant of that expression. For example when expression a1b2 – a2 b1 is expressed
in the form
a1 b1
a2 b2
,
then it is called a determinant of order 2, Clearly a determinant of order 2 contains 2 rows and 2 columns.
Similarly
a1 b1 c1
a2 b2 c2
is a determinant of order 3.
a3 b3 c3
Obviously in every determinant, the number of rows and columns are equal and this number is called the order of
that determinant.

REPRESENTATION OF A DETERMINANT
Generally we use  or |A| symbols to express a determinant and a determinant of order 3 is represented by
a11 a12 a13
a21 a22 a23
a31 a32 a33
It should be noted that the ( i, j)th element ( i.e., the element of the ith row and jth column) of the determinant has
been expressed by aij , i = 1 , 2 , 3 ; j = 1,2,3 . The elements for which i = j are called diagonal elements and the
diagonal containing them is called principal diagonal or simply diagonal of the determinant. For the above determinant
a11 , a22 , a33 are diagonal elements.
A determinant is called a triangular determinant if its every element above or below the diagonal is zero. For
example

a 0 0
b c 0
d e f

is a triangular determinant. In particular when all the elements except diagonal elements are zero, then it is called
a diagonal determinant. For example

a 0 0
0 b 0
0 0 c

is a diagonal determinant.
We generally use R1, R2, R3, ........ to denote first, second, third .... row and C1, C2 , C3 ......... to denote first,
second, third ..... column of a determinant.

VALUE OF A DETERMINANT
The expression which has been expressed in a determinant form is called the value of that determinant.
To find the value of a third order determinant
Matrices and Determinants [9]

a11 a12 a13

Let   a 21 a 22 a 23
a 31 a 32 a 33

be a third order determinant. To find its value we expand it by any row or column as the sum of three determinants
of order 2. If we expand it by first row then
a 22 a 23 a a 23 a a 22
  ( 1)11a11  ( 1)1 2 a12 21  (1)13 21
a 32 a 33 a 31 a33 a 31 a 32

a 22 a 23 a a 23 a 21 a 22
 a11  a12 21 
a 32 a 33 a 31 a 33 a 31 a 32

MINOR AND COFACTOR OF AN ELEMENT

MINOR OF AN ELEMENT
Minor of an element of the determinants is obtain by leaving the row and column containing that element and
retaining rest of elements.

a11 a12 a13 a22 a23 a21 a23

If   a21 a22 a23 , then minor of a11 is M11 = a a33
. Similarly M12 = a a
32 31 33
a31 a32 a33

Using this concept the value of Determinant can be

 = a11 M11 – a12 M12 + a13 M13
or  = – a21 M21 + a22 M22 – a23 M23
or  = a31 M31 – a32 M32 + a33 M33
COFACTOR OF AN ELEMENT
The cofactor of an element ai j is denoted by Ci j and is equal to (–1) i + j Mi j where Mi j is a minor of element ai j

a11 a12 a13

If   a21 a22 a23
a31 a32 a33

a22 a23
then C11 = (–1)1 + 1 M11 = M11 = a a33
32

a21 a23
C12 = (–1)1 + 2 M12 = –M12 = – a a
31 33

Note :- (i) The sum of products of the element of any row with their corresponding cofactor is equal to the value
of determinant i.e.  = a11 C11 + a12 C12 + a13 C13
(ii) The sum of the product of element of any row with corresponding cofactor of another row is equal to
zero i.e. a11 C21 + a12 C22 + a13 C23 = 0
(iii) If order of a determinant (  ) is 'n' then the value of the determinant formed by replacing every
element by its cofactor is  n–1
[10] Matrices and Determinants

PROPERTIES OF DETERMINANTS
If the elements of a determinant are complicated expressions or numbers, then it is very difficult to find its value
by expansion method. In such cases we reduce the determinant into a simple one using the following properties.
P–1 The value of a determinant is unchanged if its rows and columns are interchanged. For example

a b c a p u
p q r  b q v
u v w c r w

P–2 The interchange of any two consecutive rows or columns will simply change the sign of the value of the
determinant. For example

a b c b a c p q r
p q r  q p r  a b c
u v w v u w u v w

P–3 If any two rows or columns of a determinant are identical then its value is zero. For example

a b c a a b
a b c 0  p p q
u v w u u v

P–4 If each element of a row or column of a determinant be multiplied by a number, then its value is also multiplied
by that number. For example

ka kb kc a b c ka b c
p q r k p q r  kp q r
u v w u v w ku v w

P–5 If each entry in a row or column of a determinant is the sum of two numbers, then the determinant can be written
as the sum of two determinants. For example

a   b  c   a b c    a b c a b c  b c
p q r  p q r  p q r p q r  p q r   q r
and
u v w u v w u v w u  v w u v w  v w

P–6 The value of a determinant does not change if the elements of a row ( column) are added to or subtracted from
the corresponding elements of another row ( column). For example

a b c a  b   c b c
p q r  p  q   r q r
u v w u  v   w v w

P–7 If  = f (x) and f(a) = 0, then (x –a) is a factor of  . For example in the determinant
1 1 1 1 1 1
a b c b b c =0
 = if we replace a by b then  =
a2 b2 c2 b2 b2 c2
 ( a– b) is a factor of  .

P–8 If each entry in any row (or column) of a determinant is zero, then the value of determinant is equal to zero.
Matrices and Determinants [11]

MULTIPLICATION OF TWO DETERMINANTS

Multipication of two second order determinants is defined as follows
a1 b1 l1 m1 a1l 1  b1l 2 a1m1  b1m2
 
a2 b2 l2 m2 a2 l 1  b2 l 2 a2m1  b 2m2
Multiplication of two third order determinants is defined as follows
a1 b1 c1 l1 m1 n1 a1l1  b1l 2  c1l 3 a1m1  b1m2  c1m3 a1n1  b1n2  c1n3
a2 b2 c 2  l 2 m 2 n2  a2 l1  b2 l 2  c 2 l 3 a2m1  b2m2  c 2m3 a2n1  b2n2  c 3 n3
a3 b3 c3 l3 m3 n3 a3 l1  b3 l 2  c 3 l 3 a3 m1  b3m2  c 3m3 a3 n1  b3 n2  c 3 n3

Note : In above case the order of Determinant is same, if the order is different then for their multiplication first
of all they should be expressed in the same order

SYMMETRIC & SKEW SYMMETRIC DETERMINANT

Symmetric determinant
A determinant is called symmetric Determinant if for its every element.
ai j = a j i  i , j
Skew Symmetric determinant
A determinant is called skew Symmetric determinant if for its every element
a i j = – a j i  i, j ;
Note : (i) Every diagonal element of a skew symmetric determinant is always zero
(ii) The value of a skew symmetric determinant of even order is always a perfect square and that of
odd order is always zero.
i.e. (order = 2) i.e. (order = 3)

APPLICATIONS OF DETERMINANT
CRAMMER'S RULE :
Let the system of equations be
a1 x + b1 y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
a1 b1 c1 d1 b1 c1 a1 d1 c1 a1 b1 d1
  a2 b2 c2
and , 1  d2 b2 c 2  2  a2 d2 c2
, 3  a2 b2 d2
a3 b3 c3 d3 b3 c3 a3 d3 c3 a3 b3 d3
then (i) If   0 , then given system of equations is consistent i.e. has unique solution and its solution is
1  
x , y  2 , z 3
  
This is known as Crammer's rule
(ii) If  = 0 and atleast one of  1 ,  2 ,  3 is not zero, then the system of equations is inconsistent i.e. it
has no solution.
(iii) If  = 0 and  1 = 0 =  2 =  3 , then the system has infinite solutions.
(iv) If  = 0 and d1 = 0 = d2 = d3 , then the system of equations has infinite solutions ( non-zero solution) i.e.
non-trivial solutions
[12] Matrices and Determinants

(v) If   0 and d1 = 0 = d2 = d3, then the system of equations has a unique solution x = 0 , y = 0 , z = 0 i.e.,
zero solution or trivial solution.

DIFFERENTIATION OF A DETERMINANT :

f1 (x) g1 (x) h1 (x)

Let (x)  f 2 (x) g 2 (x) h 2 (x) ,
f 3 (x) g3 (x) h 3 (x)
f1 '(x) g '1 (x) h '1 (x) f1 (x) g1 (x) h1 (x) f1 (x) g1 (x) h1 (x)
then  '(x)  f 2 (x) g 2 (x) h 2 (x)  f '2 (x) g '2 (x) h '2 (x)  f 2 (x) g 2 (x) h 2 (x)
f 3 (x) g 3 (x) h 3 (x) f3 (x) g3 (x) h 3 (x) f '3 (x) g '3 (x) h '3 (x)

INTEGRATION OF DETERMINATION

f (x) g(x) h(x)

Let (x)  p q r , where p, q, r, l, m and n are constants.
l m n

b b b

b 
a
f (x)dx 
a
g(x)dx 
a
h(x)dx

Then a
(x)dx  p q r
l m n

USE OF SUMMATION

n n n
2 3
r r 2 3
r n
r r r
r 1 r 1 r 1
If f (r)  p q
1 2 3
t , where p, q, t are constants, then  f (r) 
r 1
p
1
q
2
t
3