Matrices and Determinants chapter from TN syllabus

1. STANDARD 11th MATHEMATICS
VOLUME-II CHAPTER 7
MATRICES AND DETERMINANTS
1
By
LILITHA R

2. 2
INTRODUCTION
MATRICES
DETERMINANTS
SUMMARY
CONTENT
 Types of Matrices
 Equality of Matrices
 Algebraic operations on Matrices
 Symmetric and Skew-symmetric Matrices
 Determinants of matrices of different order
 Properties of Determinants
 Product of Determinants

3. 3
Matrices
A mat rix is a rect ang ular array or arrang ement
of ent ries or element s displayed in rows and
columns put within a square bracket [ ]. In
g eneral, t he ent ries of a mat rix may be real or
complex numbers or funct ions of one variable
(such as polynomials, t rig onomet ric funct ions or
a combinat ion of t hem) or more variables or
any other object. Usually, matrices are
denot ed by capit al let t ers A, B, C, ... et c. In
t his chapt er t he ent ries of mat rices are
rest rict ed t o eit her real numbers or real
valued funct ions on real variables.

4. 4
2 3 7
1 1 5
A
 
  

 
1 3 1
2 1 4
4 7 6
B
 
 
  
 
 
Both A and B are examples of matrix. A matrix
is a rectangular array of numbers enclosed by a
pair of bracket.
Why matrix?

5. 5
How about solving
7,
3 5.
 


 

x y
x y
2 7,
2 4 2,
5 4 10 1,
3 6 5.
  
  
   
 





 

x y z
x y z
x y z
x y z
Consider the following set of equations:
It is easy to show that x = 3 and
y = 4.
Matrices can help…
Matrices

6. 6
11 12 1
21 22 2
1 2
 
 
 

 
 
 
n
n
m m mn
a a a
a a a
A
a a a
In the matrix
numbers aij are called elements. First subscript
indicates the row; second subscript indicates
the column. The matrix consists of mn elements
It is called “the m  n matrix A = [aij]” or simply
“the matrix A ” if number of rows and columns
are understood.
Matrices

7. 7

8. 8

9. 9
Equal matrices
Two matrices A = [aij] and B = [bij] are said to
be equal (A = B) iff each element of A is equal
to the corresponding element of B, i.e., aij = bij
for 1  i  m, 1  j  n.
iff pronouns “if and only if”
if A = B, it implies aij = bij for 1  i  m, 1  j  n;
if aij = bij for 1  i  m, 1  j  n, it implies A = B.
Matrices

10. 10
Equal matrices
Given that A = B, find a, b, c and d.
Matrices
1 0
4 2
A
 
  

 
a b
B
c d
 
  
 
Example: and
if A = B, then a = 1, b = 0, c = -4 and d = 2.

11. 11

12. 12

13. 13

14. 14

15. 15
Symmetric & Skew Symmetric matrix

16. Determinants
If is a square matrix of order 1,
then |A| = | a11 | = a11
ij
A= a
 
 
If is a square matrix of order 2, then
11 12
21 22
a a
A =
a a
 
 
 
|A| = = a11a22 – a21a12
a a
a a
1
1 1
2
2
1 2
2

17. Properties of Determinants
1. The value of a determinant remains unchanged, if
its rows and columns are interchanged.
1 1 1 1 2 3
2 2 2 1 2 3
3 3 3 1 2 3
a b c a a a
a b c = b b b
a b c c c c
i e A A

. . '
2. If any two rows (or columns) of a determinant are
interchanged, then the value of the determinant is changed
by minus sign.
 
1 1 1 2 2 2
2 2 2 1 1 1 2 1
3 3 3 3 3 3
a b c a b c
a b c = - a b c R R
a b c a b c
Applying 

18. 3. If any two rows (or columns) of a determinant
are identical, then its value is zero.
2 2 2
3 3 3
0 0 0
a b c =0
a b c
4. If each element of a row (or column) of a determinant is
zero, then its value is zero.
1 1 1
2 2 2
1 1 1
a b c
a b c = 0
a b c

19. 19
Products of Determinants

20. Example

21. THANK YOU

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