Matrices

Presented by

Ajay Gupta

AJAY GUPTA PGT MATHS
CONT. NO. 9868423152

KV VIKASPURI,
NEW DELHI

MATRICES
 A matrix is a rectangular array
(arrangement) of numbers real
or imaginary or functions kept
inside braces () or [ ]subject to
certain rules of operations.
3 2 3
4 5 
3


2 4 5


ORDER OF A MATRIX
 A matrix having ‘m’ number of rows
and ‘n’ number of columns is said to be
of order ‘m ×n’
I row 1 − 1 1 
II row 2 1 − 3
III row  
1
 1 1

I II III
Columns

Notation of a Matrix
1. In compact form matrix is represented by
A = [a i j ] m × n
2. The element at i th row and j th column is called the
(i, j) th element of the matrix i.e. in a i j the first
subscript i always denotes the number of row
and j denotes the number of column in which the
element occur.
3. A matrix having 2 rows and 3 columns is of order
2 × 3 and another matrix having 1 row and 2
columns is of order 1 × 2.

Location of the elements in a matrix
For matrix A

a11 a12 a13 
a a 22 
a 23 
 21
a31
 a 32 a 33 

 − 2 5 6 − 7 9

 7 4 3  6 
   5

TYPES Of MATRICES

MATRICES

ROW MATRIX

COLUMN MATRIX

SQUARE MATRIX
SQUARE MATRIX
DIAGONAL MATRIX

ZERO MATRIX
SCALAR MATRIX

SYMMTRIC MATRIX IDNTITY MATRIX

SKEW-SYMMETRIC MATRIX

ROW / COLUMN MATRICES
1. Matrix having only one row is called Row-
Matrix i.e. the row matrix is of order 1 ×
n.
2. Matrix having only one column is called Column-

[2 5 8]
matrix i.e. the column matrix is of order m × 1.
 − 5
4
 

ZERO MATRIX
A matrix whose all the elements are
zero is called zero matrix or null matrix
and is denoted by O i.e. a i j = 0 for all
i, j.

0  0 0 
0  0 0 
 
 

1. SQUARE matrix is a matrix having same
number of rows and columns and square
matrix having ‘n’ number of rows and
columns is called of order n
2. DIAGONAL matrix is a square matrix if all
its elements except in leading diagonal are
zero i. e. a ij = 0 for i ≠ j and a ij ≠ 0 for i = j.
3. SCALAR matrix is the diagonal matrix with
all the elements in leading diagonal matrix
are same i.e. a ij = 0 for i ≠ j. and a ij = k for i
= j.
4. UNIT matrix is the scalar matrix with all the
elements in leading diagonal 1 i.e. a ij = 0 for
i ≠ j. and a ij = 1 for i = j.

1 4 1 2 − 5
6 1 0   2 0 
8 9   8 9  0 −1 0 2
  7
 2 3     

 2 0 0
0 − 2 0
1 0 0
0 1 0   0 1  1 2
    1 0   
 0 0 3
  0 0 1 
    2 1
1 0 0   0 1 2  1 0 0
0 1 0   0 0 1  0 − 1 0 
     
0 0 1   2 4 0 
    0 0 1 
 

OPERATION ON MATRICES
Matrices support different basic operations .
Some of the basic operations that can be applied are
1. Addition of matrices.
2. Subtraction of matrices.
3. Multiplication of matrices.
4. Multiplication of matrix with scalar value.
But two matrices can not be divided.

EQUALITY OF MATRICES
Two matrices are EQUAL if both are of same
order and each of the corresponding element
in both the matrices is same.


1 2
 1 3 5  1 − 2  1 − 2
4 3 4 3 4
3   2 4 6    

5 6
   − 7 8 
  − 7 − 8 
 

[2 5 9] [2 5 9]

ADDITION OF MATRICES
Two or more matrices of same order can be add up to
form single matrix of same order.

 2 − 1 4  8 0 3
 − 7 5 6 +  − 1 2 4
   


 2 − 1 4  8 0 3
 − 7 5 6 +  − 1 2 4
   
2 + 8 _ _
 _ 
_ _



 2 − 1 4 8 0 3
 − 7 5 6 + − 1 
2 4
  
2 + 8 − 1 + 0 _
 _ _ 
_



 2 − 1 4 8 0 3
 − 7 5 6 + − 1 2 
4
  
 2 + 8 − 1 + 0 4 + 3
 _ _ _ 



 2 − 1 4  8 0 3
 − 7 5 6  +  − 1 2 4
   
 2 + 8 − 1 + 0 4 + 3
− 7 − 1 _ _ 



 2 − 1 4  8 0 3
 − 7 5 6  +  − 1 2 4
   
 2 + 8 − 1 + 0 4 + 3
− 7 − 1 5 + 2 _ 



 2 − 1 4  8 0 3
 − 7 5 6  +  − 1 2 4
   
 2 + 8 − 1 + 0 4 + 3
 − 7 − 1 5 + 2 6 + 4
 


 2 − 1 4 8 0 3
 − 7 5 6 + − 1 
2 4
  
 2 + 8 − 1 + 0 4 + 3  10 − 1 7 
= =  − 8 7 10
 − 7 − 1 5 + 2 6 + 4  

PROPERTIES OF MATRIX
ADDITION
 A+B=B+A
 A + ( B + C) = (A + B) + C

A+ 0 = 0 +A=A

 A + (-A) = 0 = (-A) + A

A+ B =A+ C  B = C

MULTIPLICATION OF MATRIX
WITH SCALAR
 For matrix A of order m × n and scalar number k, the
matrix of order m × n obtained by multiplying each
element of A with k is called scalar multiplication of
A by k and is denoted by kA.
1 − 2 3
 For A =  4 5 0
 
1× 2 _ _
 2A =  _ _ 
_


WITH SCALAR
1 − 2 3
 For A = 4 5 
0

2 _ _
 2A = _ _ 
_


WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
 2 2 × −2 _ 
 2A = _ _ _


WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
2 − 4 _
 2A = _ _ _ 
 

WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
 2 − 4 2 × 3
 2A =  
_ _ _ 

WITH SCALAR
1 − 2 3
 For A =  
 4 5 0
2 − 4 6
 2A =  
_ _ _ 

WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
 2 − 4 6
 2A = 2 × 4 _ _
 

WITH SCALAR
1 − 2 3
 For A =  
4 5 0
2 − 4 6
 2A =  
8 _ _

WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
2 − 4 6 
 2A = 8 2 × 5 _ 
 

WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
2 −4 6
 2A = 8 10 
_


WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
2 − 4 6 
 2A = 8 10 2 × 0
 

WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
2 −4 6
 2A = 8 10 
0


WITH SCALAR
1 − 2 3
 For A =  
4 5 0
− 3 _ _
 -3A =  
_ _ _

WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
− 3 6 _
 -3A = _ _ 
_


WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
 − 3 2 − 9
 -3A =  _ _ _ 


WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
 − 3 2 − 3
 -3A = − 12 _ _ 
 

WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
 −3 2 − 3
 -3A =  − 12 − 15 _ 
 

WITH SCALAR
1 − 2 3
 For A =  4 5 0
 
 − 3 2 − 3
 -3A =  − 12 − 15 0 
 

PROPERTIES OF SCALAR
MULTIPLICATION
 k (A + B) = k A + k B

 (-k) A = - (k A) = k (-A)

 IA=AI=A

 (-1) A = - A

MULTIPLICATION OF MATRICES
 Two matrices can be multiplied only if number
of columns of first is same as number of rows
of the second.
 If A is of order m × n and B is of order n × p,
then the product AB is a matrix of order m × p.
 m × n & n × p  m × p.
 For A = [a i j] m×n and B = [ b j k] n×p , AB = C with
C = [cij] m×p where ci k = Σ a ij b jk


 6 9   2 6 0
  
 2 3  7 8 9 =
 


 6 9  2 6 0  6 .2 + 9 .7 _ _
   =  
 7 8 9  
 2 3    _ _ _


 6 9  2 6 0  6.2 + 9.7 6.6 + 9.8 _ 
   = 
 
 7 8 9
 _ _ _
 2 3  


 6 9   2 6 0   6.2 + 9.7 6.6 + 9.8 6.0 + 9.9 
  7 8 9 = 
    
 2 3    _ _ _  


 6 9  2 6 0  6.2 + 9.7 6.6 + 9.8 6.0 + 9.9 
   
 7 8 9 = 
 2.2 + 3.7 

 2 3    _ _ 


 6 9  2 6 0  6. 2 + 9. 7 6. 6 + 9. 8 6. 0 + 9. 9 
   
 7 8 9 = 
 2.2 + 3.7 2.6 + 3.8 _ 

 2 3    


 6 9  2 6 0  6.2 + 9.7 6.6 + 9.8 6.0 + 9.9 
  
 7 8 9
 = 
 2.2 + 3.7 2.6 + 3.8 2.0 + 3.9 

 2 3    


6 9   2 6 0   6.2 + 9.7 6.6 + 9.9 6.0 + 9.8 
  ×   =  
2 3   7 9 8   2.2 + 3.7 2.6 + 3.9 2.0 + 3.8 


 6.2 + 9.7 6.6 + 9.9 6.0 + 9.8 
 =

 2.2 + 3.7 2.6 + 3.9 2.0 + 3.8 
 
 75 117 72 
= 
 25 30



24 

TRANPOSE OF MATRIX

 For matrix A = [aij] of order m×n,
/
transpose of A is denoted by A of A and
T

it is a matrix of order n×m and is
obtained by interchanging the rows with
columns i.e. AT=[aji] with aij = aji for all i,j.

 1 5
 9 8 A  1 9 − 1
A=  
T
= 
 − 1 3 3× 2  5 8 3  2× 3
 

PROPERTIES OF
TRANSPOSE OF MATRICES
 (AT)T = A
 (A + B)T= AT + BT
 (kA)T = k AT
 (AB)T = BT AT
 Every square matrix can be
expressed as sum of sum of
symmetric and skew-symmetric
1 (A + AT) + 1(A – AT)
matrix. A = 2
2

SYMMETRIC/SKEW-SYMMETRIC
MATRICES
 A square matrix A = [aij] is called symmetric
matrix if AT = A i.e. aij = aji for all i,j.
 A square matrix A = [aij] is called skew-
symmetric matrix if AT = -A i.e. aij = - aji for
all i,j.
 1 2 − 3  0 −2 3 
2 0 7 2 
0 − 7
  
 − 3 7 − 2 − 3 7
 0
 

IMPORTANT RESULT ON SYMMETRIC
AND SKEW-SYMMETRIC MATRICES

 Everysquare matrix can be expressed as
sum of sum of symmetric and skew-
symmetric matrix. A =(A + AT) +(A – AT)

 Allthe elements in lead diagonal in skew-
symmetric matrix are zero.

APPLICATION OF MATRICES

 Solution of equations in AX=B system
using matrix method
−1
(i) If A = 0 unique solution with X = A B
/
(ii) If A = 0, and also (adjA)B = 0, Infinite
many solutions.
(iii) If A = 0, (adj A) B = 0 No solution.
/

Important Problems
1 Construct a 2 3 matrix A with elements given by
i +2 j
aij =
i−j
2 Find x, y such that  x − y 2 − 2 3 − 2 2  6 0 0
 4  + 1 0 − 1 = 5 2 x + y 5
x 6  
   

3 If A = diag.(2 -5 9), B = diag.(1 1 -4), find
3A – 2B.
3 2 1 0
4 Find X and Y if 2X + Y = 1 4 and X + 2Y = 
   − 3 2


1 3 2  
1
5 Find x if [
1 x 1] 
2 5 1   0
2
 =

15
 3 2  
x

 3 1
6 If A =  − 1 2 show that A 2 − 5 A + 7 = 0
 
.  1



w w2   w
 
w2 
1 1





 w
2
w 1 + w 2
 1 w  w
 =0
7 Show that 

 w2
 1 w  1
  w 2  
 w
w  2



 3 1
8 If A = − 1 2 and Find K so that A 2 = 5 A + KI
 
.
9 Show that B ′AB is symmetric or skew-
symmetric according as A is symmetric or
skew-symmetric.

10 Express A = 3 2 as sum of symmetric
3
4 5 3
 

2 4 5


and skew-symmetric matrices.
11 If A = 3 2 find
( AB )
4 6 −1
2 5  & B −1 = 
   3 2

12 Find X if
3 2  − 1 1  2 − 1
7 5 X − 2 1 = − 1 4 
     
13 Solve using matrix method
x + 2y + z = 7, x + 3 z = 11, 2 x – 3 y = 1.

Address of the subject related websites

 http://www.netsoc.tcd.ie/~jgilbert/maths_site/apple

 http://www.ping.be/~ping1339/matr.htm

 http://mathworld.wolfram.com/Matrix.html

 http://en.wikipedia.org/wiki/Matrices

ACKNOWLEDGEMENT

This power point presentation is prepared
under the active guidance of Ms. Summy and
Ms. Nidhi the able and learned trainers of
project “SHIKSHA” CONDUCTED BY
MICROSOFT CORPORATION.

Matrices

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