Matrices & Determinants (Solved MCQs and Notes)

Notes 60 Pages

Joint Entrance Examination (Main)

Mathematics

Contributed by

Nikita Narasimhan

124
Unit - 3
Matrices and Determinants
Important Points
• Matrix :
Any rectangular array of numbers is called matrix. A matrix of order m  n having m
rows and n columns. Its element in the i
th
row and j
th
column is a
ij
. We denote matrix
by A, B, C etc.
1 2
3 0
 
 
 
is a matrix of order 2  2.
1 0 2
3 2 1

 
 
 
is matrix of order 2  3.
A matrix of order m  n is
11 12 1n
21 22 2n
m1 m2 mn
 
 
 
 
 
 


  

a a a
a a a
a a a
• Algebra of Matrices
(1) Equality : If
m n p q
A , B b
 
   
 
   
ij ij
a
are said to be equal i.e. A = B if
(i) a
ij
= b
ij
 i & j
(ii) order of A = order of B, i.e. m = p and n = q
• Types of Matrices : Let
m n
A

 

 
ij
a
(1) Row matrix : A 1  n matrix [a
11
a
12
a
13
..... a
1n
] is called a row matrix
(row vector)
(2) Column matrix : A m  1 matrix
11
21
31
n1
 
 
 
 
 
 
 
 

a
a
a
a
is called column matrix
(Column vector)
Page 1
125
(3) Square Matrix : An n  n matrix is called a square matrix.
(4) Diagonal matrix : If in a square matrix A = [a
ij
]
n  n
we have a
ij
= 0 whenever i

j then A is called a diagonal matrix.
(5) Zero (null) matrix : A matrix with all elements are zero is called zero (null)
matrix. It is denoted by [0]
m x n
or O
m x n
or O.
• Algebra of Matrices
(2) Sum and Difference : If A and B are of same oder
i.e.
n n
A , B ,C
  
     
  
     
ij ij ij
m n m m
a b c
then A + B = C
n
+
ij ij ij
m m n
a b c
 
   
 
   
A - B = C
n
ij ij ij
m m n
a b c
 
   
  
   
Properties of addition
If matrices A, B, C and O are 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 (Existence of Identity)
(iv) (-A) + A = A + (-A) = O (Existence of Inverse)
(3) Product of Matrix with a Scalar
If A =

ij
m n
a

 
 
and k R then we define product of matrix with a scalar is

kA k k
 
   
 
   
ij ij
m n m n
a a
Properties of Addition of Matrices and Multiplication of a Matrix by a scalar
Let
n
A , B , , R
ij ij
m n m
a b k l
 
   
  
   
(i) k(A + B) = kA + kB
(ii) (k + l) A = kA + lA
(iii) (kl)A = k(lA)
(iv) 1A = A
(v) (-1) A = -A
(4) Matrix Multiplication :
Let
n p
A , B
ij ij
m n
a b
 
   
 
   
. Then AB = C
Page 2
126
where
1 1 2 2 3 3
1
.........
n
ij ik kj i j i j i j in nj
k
C a b a b a b a b a b

     

= Scalar product of i
th
row of A and j
th
Column of B.
(i) Product AB defined if and only if number of column of A = number of
rows of B
(ii) If A is m  n matrix and B is n  p matrix then AB is m  p matrix.
• Properties of Matrix Multiplication
Let the matrices A, B, C and O have order compitable for the operations involved.
(i) A(B + C) = AB + AC
(ii) (A + B)C = AC + BC
(iii) A(BC) = (AB) C
(iv) AO = O = OA
(v) AB  BA, generally
(vi) AB = O need not imply A = O or B = O
(vii) AB = AC need not imply B = C
• Types of Matrices
(6) Identity (unit) Matrix : In a diagonal matrix all elements of principal
diagonal are 1 is called Identity (unit) Matrix and is denoted by I or I
n
or I
n  n.
(7) Scalar Matrix : If k R, then kI called a scalar matrix.
(8) Traspose of a Matrix : If all the rows of matrix

A
ij
m n
a

 

 
are converted
into corresponding column, the matrix so obtained is called the transpose of
A. It is denoted by A
T
or
A '
. A
T
=
n m
 
 
ji
a
Properties of Transpose
(i) (A
T
)
T
= A
(ii) (A + B)
T
= A
T
+ B
T
(iii) (kA)
T
= kA
T
, k  R
(iv) (AB)
T
= B
T
A
T
(9) Symmetric Matrix : For a square matrix A, if A
T
= A, then A is called a
symmetric matrix. Here a
ij
= a
ji
for all i and j.
(10) Skew - Symmetric Matrix : For a square matrix A, if A
T
= -A, then A is called
a
Skew - symemtric matrix.
Here a
ij
= -a
ji
for all i and j and a
ii
= 0

i
Page 3
127
For square matrix A, A + A
T
is symmetric and A - A
T
is skew - symmetric
matrix.
(11) Triangular Matrices :
(i) Upper Triangular Matrix : A sqare matrix whose element a
ij
= 0 for i >
j is
called an uppear triangular matrix.
e.g.
b
, 0
0 c
0 0
 
 
 
 
 
 
 
 
a b c
a
d e
f
(ii) Lower Triangular Matrix : A square matrix whose element a
ij
= 0 for i
< j is
called a lower triangular matrix.
e.g.
0 0
0
, 0
b c
 
 
 
 
 
 
 
 
a
a
b c
d e f
Let A be a square matrix of order n  n.
(12) Orthogonal matrix : A is called an orthogonal matrix if and only if A
T
A = I
n
= A A
T
(13) Idempotent Matrix : A is called an idempotgent matrix if A
2
= A
(14) Nilpotent Matrix : A is called a nilpotent matrix if A
m
= 0, m Z
+
(15) Involutary Matrix : A is called an involutary matrix if A
2
= I, i.e. (I + A) (I -
A) = O
(16) Conjugate of a Matrix : If A = [a
ij
] is a given matrix, then the matrix obtained
on replacing all its elements by their corresponding complex cojugates is
called the conjugate of the matrix A and is denoted by
A
 

 
ij
a
Properties :
(i)
(A) A
(ii)
(A + B) A + B
(iii)
(kA) k A, k being a complex number 
(iv)
AB A B 
(17) Conjugate Transpose of a matrix : The conjugate of the transpose of a given
matrix A is called the conjugate transpoe (Tranjugate) of A and is denoted
by
θ
A
.
Page 4
128
Properties :
(i)


 
T
θ T
A    
(ii)
 
θ
θ
A A
(iii)
 
θ
θ θ
A + B + B 
(iv)
 
θ
θ
kA k A , k being a complex number 
(v)
 
θ
θ θ
AB B A 
Let A be a square matrix of oder n  n
(18) Unitary Matrix : A is an unitary matrix if AA

= I
n
= A

A.
(19) Hermitian Matrix : A is a hermitian matrix if A

= A
(20) Skew - Hermitian Matrix : A is a skw-Hermitian matrix if A

= -A
• The determinant of a square matrix :
If all entries of a square matrix are kept in their respective places and the
determinant of this array is taken, then the detrminant so obtained is called the
determinant of the given square matrix. If A is a square matrix, then determinant of
A is denoted by | A | or det A.
Evaluation of Determinants (Expansion)
Second order determinant
b
c d
a
ad bc 
Third order determinant
1 1 1
2 2 2 2 2 2
2 2 2 1 1 1
3 3 3 3 3 3
3 3 3
b c
b c c b
b c b c
b c c b
b c
a
a a
a a
a a
a
  
= a
1
(b
2
c
3
- b
3
c
2
) - b
1
(a
2
c
3
- a
3
c
2
) + c
1
(a
2
b
3
- a
3
b
2
)
= a
1
b
2
c
3
- a
1
b
3
c
2
- a
2
b
1
c
3
+ a
3
b
1
c
2
+ a
2
b
3
c
1
- a
3
b
2
c
1
Some Symbols :
(1) R
i
 C
i
: To convert every row (column) into corresponding column (row)
(2) R
ij
[c
ij
] (i  j) : Interchange of i
th
row (column) and j
th
row (column)
(3) R
i
(k) [c
i
(k)] : multiply i
th
row (Column) by kR - {0}
(4) R
ij
(k)[c
ij
(k)] : Multiply i
th
row (column) by kR (k  0) and adding to the
corresponding elements of j
th
row(column)
Page 5
129
Properties of Determinants of Matrices
(i) | A
T
| = | A |
(ii) | AB | = | A | | B |
| A B C | = | A | | B | | C |
(iii) | kA | = k
n
| A | (where A is n  n matrix)
(iv) | I | = 1
Value of some Determinants :
(i) Symmetric Determinant
2 2 2
p q
p y r yz + 2pqr yq r
q r z
   
x
x xp z
(ii) Skew - symmetric determinant of odd order
0 y
: 0 z
y z 0
  

x
x
(iii) Circular Determinant
 
3 3 3
y z
: y z y z 3
z y
    
x
x x xyz
x
Area of a Triangle :
If the vertices of a triangle are (x
1
, y
1
), (x
2
, y
2
) and (x
3
, y
3
) then,
Area of a triangle =
1 1
2 2
3 3
y 1
D , where D y 1
y 1
x
x
x

    

Shifting of origin does not effect the area of a triangle.
If D = 0  all three points are collinear
Let the sides of the triangle be a
1
x + b
1
y + c
1
= 0, a
2
x + b
2
y + c
2
= 0, a
3
x + b
3
y + c
3
= 0
1 2 3
Area of a triangle ,
2 C C C


 
 
where C
1
, C
2
, C
3
are respetively the cofactors of c
1
, c
2
and c
3
and
1 1 1
2 2 2
3 3 3
b c
b c
b c
a
a
a
 
Page 6
130
If A(x
1
, y
1
) and B(x
2
, y
2
) are two distinct points of
AB

then the cartesian equation of
AB

is
1 1
2 2
1
y 1
y 1
x y
x
x
 
Propeties of Determiants : (D = value or determinante)
(i) If a row (column) is a zero vector (i.e. all elements of a row or a column are
zero), then D = 0
(ii) If two rows (Columns) are identical, then D = 0
(iii) If any two rows (columns) are interchanged, then D becoms - D (additive
i n v e r s e
of D)
(iv) If any two rows (columns) are interchanged, D is unchaged  | A
T
| = | A |
(v)
1 1 1 1 1 1
2 2 2 2 2 2
3 3 3 3 3 3
k kb kc b c
b c k b c
b c b c
a a
a a
a a

(vi)
1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3
+ d b + e c + f b c d e f
b c b c + b c
b c b c b c

a a
a a a
a a a
(vii) If any rows (columns) is multiplied by k R (k  0) and added to another rows
(columns), then D is unchanged.
1 1 1 1 2 1 2 1 2
2 2 2 2 2 2
3 3 3 3 3 3
b c + k b + kb c + kc
b c b c
b c b c
a a a
a a
a a

(vii) All rows of a determinant are converted into corresponding column, D is
unchanged.
(viii) Determinants are multiplied in the same way as we multiply matrices.
T T T T
|AB| A B | BA | = AB | | A B = A B| |        
(ix)
1 1 1
2 2 2 r r r
3 3 3
f g h
f g h , where f , g , h are functions of for r , 2, 3.
f g h
  x
Page 7
131
1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3
f ' g ' h ' f g h f g h
d
f g h + f ' g ' h ' + f g h
d
f g h f g h f ' g ' h '
x

 
(x) Let D(x) be a 3  3 determinant whose elements are polynomials.
If D(m) has two identical rows (columns), then x - m is a factor of D(x)
If D(m) has three identical rows (columns), then (x - m)
2
is a factor of D(x).
• Minor and cofactor
Let
11 12 13
21 22 23
3 3
31 32 33
A

 
 
 
 
 
 
 
ij
a a a
a a a a
a a a
The minor of the element a
ij
(i, j = 1, 2, 3) in A
= M
ij
= The determiant obtained from A on deleting the row and the column in
which a
ij
occurs.
The cofactor of the element aij (i, j = 1, 2, 3) in A = A
ij
= (-1)
i + j

M
ij
The value of any third order determinant can be obtained by adding the products of
the elements of any of its rows (columns) by their correspdong co-factor.
If we multiply all the elements of any rows (columns) of any third order
determinant by the cofactos of the corresponding elements of another row
(column) and add the products, then the sum is zero.
or in Mathematical notation

j
A if i = k = 1, 2, 3
0 if i k = 1, 2, 3
ij kj
a


 



i
A if j = k = 1, 2, 3
0 if j k = 1, 2, 3
ij ik
a


 


• Adjoint of Matrix
Adjoint Matrix of A = adj A
11 21 31
12 22 32
13 23 33
a a a
a a a
a a a
 
 

 
 
= Tranpose of the matrix of cofactor = [A
ji
]
3  3
If A = [a
ij
]
n  n
then adj A = [A
ji
]
n  n
To obtain the adjoint of 2  2 matrix, interchange the elements on the principal
diagonal and change the sign of the elements on the secondary diagonal.
Properties of Adjoint Matrix : If A is square matrix of order n,
(i) A(adj A) = (adj A) A = | A | I
n
(ii) adj I
n
= I
n
(iii) adj (kI
n
) = k
n - 1
I
n
, k is a scalar.
Page 8
132
(iv) adj A
T
= (adj A)
T
(v) adj (kA) = k
n - 1
adjA, k is a scalar.
(vi) adj(AB) = (adj B)(adj A)
(vii) adj (ABC) = (adj C)(adj B)(adj A)
If A is a non singular matrix of order n, then
(i) | adj A | = | A |
n - 1
(ii) adj (adj A) = | A |
n - 2
A
(iii) | adj (adj A) | = | A |
(n - 1)
2
Adjoint of
(i) a diagonal matrix is diagonal
(ii) a triangular matrix is triangular
(iii) a symmetric matrix is symmetric
(iv) a hermitian matrix is thermitian
• Inverse of a Matrix
A square matrix A is said to be singular if | A | = 0 and non singular if | A | = 0
฀
If A is a square matrix of order n, if there exists another square matrix of order n
such that
A B = I
n
= BA
Then B(A) is called inverse of A(B).It is denoted A
-1
.

1
A A)
| A |
adj


 
If inverse of matrix A exists, then it is unique.
A square matrix A is non-singular
| A |   

1
A exists.


Results :
(i)
1 1
A
 
   
(ii)
1 1 1
(AB) A
  
 
(iii)
T 1 1 T
(A ) (A )
 

(iv)
k 1 1 k
(A ) (A ) , k Z
 
 
(v) A = diag [a
11
a
22
a
33
..... a
nn
] and a
11

a
22

a
33


...

a
nn
 0 then
A
-1
= diag [a
11
-1
a
22
-1
a
33
-1
.......... a
nn
-1
]
(vi) Inverse of a symmetric matrix is symmetric.
• Elementary Transformations (operations) of a matrix
(i) Interchange of rows (columns)
Page 9
133
(ii) The multiplication of the elements of a row (column) by a non - zero scalar.
(iii) The addition (subtraction) to the elements of any row (column) of the scalar
multiples of the corresponding elements of any other row (column).
• Test of Consistency
If the system of equation possesses atleast one solution set (solution set is not
empty) then the equations are said to be consistent.
If the system of equation has no solution they are said to inconsistent.
Solution of simultaneous linear equations in two (three) variables :
Trival solution :
V alue of all the vari abl es is zero i .e.
x = 0, y = 0, z = 0
Non Triavial Solution :
Value of atleast one variable is non-zero
Homogeneous linear equation :
If constant term is zero, i.e. ax + by = 0 or ax + by + cz =0
such equations is called homogenous linear eqaution.
Solutiuon of homogenoeous linear equation
Consider the eqautions
For three variables For two variables
a
11
x + a
12
y + a
13
z = 0 a
11
x + a
12
y = 0
a
21
x + a
22
y + a
23
z = 0 a
21
x + a
22
y = 0
a
31
x + a
32
y + a
33
z = 0
11 12 13
21 22 23
31 32 33
y
z
a a a
x
a a a
a a a

 
   
 
   
 
 
   

   
 
11 12
21 22

y
a a
x
a a

     

 
   

    
A X = O A X = O
(i) If | A |  0 the system is consistent and has only trivial (unique) solution.
(ii) If | A | = 0 the system is consistent and has non trivial (infinite number of)
Solution.
• Solution of non-homogeneous linear equation :
Let three equations a
1
x + b
1
y + c
1
z = d
1
a
2
x + b
2
y + c
2
z = d
2
a
3
x + b
3
y + c
3
z = d
3
Page 10

Matrices & Determinants (Solved MCQs and Notes)

Matrices & Determinants (Solved MCQs and Notes)

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