The document discusses various topics related to matrices including determinants, Cramer's rule, and applications of matrices. It provides definitions and examples of determinants, properties of determinants, calculating a 2x2 determinant, and Cramer's rule for 2x2 and 3x3 matrices. It also demonstrates finding the inverse of a matrix using the adjoint method and provides an example of using matrices to solve a system of linear equations.

1. ROLL NO. NAME TOPICS DISTRIBUED
078 BHARGAVI THATI DETERMINANTS
004 PREETHI ALLE CRAMERS RULE
011 TANU BHARDWAJ
INVERSE BY ADJOINT
METHOD
072 JAYALAXMI SHETTY APPLICATIONS OF
MATRICES

2. DETERMINANTS

3. Determinant is defined for square matrices. A square matrix has the same number of rows as
columns, like a matrix or a matrix .
We will discuss few applications of determinant. The cross product of two vectors in can be
defined using the determinant of a matrix. The area of the parallelogram generated by these
two vectors can be be obtained using as a determinant. The volume of the parallelepiped formed
by any three nonzero vectors in , also can be find using determinant. Determinant is used in
change of variable of integrals in calculus. It can be used in finding eigenvalues of matrix.
Determinant of a matrix can tell us about invertible of the matrix, number of solutions of an a
linear system of n-equations in n-unknowns, and many other applications.
INTRODUCTION

4. Here are some of the important properties of determinants which can make easier
calculations:
1. Reflection Property:
The reflection property of Determinants defines that Determinants do not change if rows are transformed into
columns and columns are transformed into rows.
2. All- Zero Property:
The Determinants will be equivalent to zero if each term of rows and columns are zero.
3. Proportionality (Repetition Property):
If each term of rows or columns is similar to the column of some other row (or column) then the Determinant
is equivalent to zero.
4. Switching Property:
The interchanging of any two rows (or columns) of the Determinant changes its signs.
IMPORTANT PROPERTIES OF DETERMINANTS

5. 5. Factor Property:
If a determinant ▲ becomes zero when we put x=a, then (x-a) is a factor of ▲.
6. Scalar Multiple Property:
If all the elements of a row (or columns) of a Determinant are multiplied by a non-zero
constant, then the Determinant gets multiplied by a similar constant.
7. Sum Property:
8. Triangle Property:

6. 9. Determinant of Cofactor Matrix:
10. Property of Invariance:

7. A 2 × 2 Determinant
In general, we find the value of a 2 × 2
determinant with elements a, b, c, d as
follows:

8. Calculating a 2 × 2 Determinant
We multiply the diagonals (top left ×
bottom right first), then subtract.

9. Cramer's Rule
Preethi Alle

10. Definition
Cramer's Rule is an explicit formula
for the solution of a system of linear
equations with as many equations as
unknowns, i.e. a square matrix, valid
whenever the system has a unique
solution.

11. • Cramer’s Rule is efficient for solving small systems and
can be calculated quite quickly; however, as the system
grows, calculating the new determinants can be tedious.
Key Points
• Cramer’s Rule only works on
square matrices that have a non-
zero determinant and a unique solution.

12. Cramer's rule for
2*2

13. Cramer's rule for
3*3

14. INVERSE OF MATRIX
BY ADJOINT METHOD

15. INVERSE BY ADJOINT METHOD:
𝐴 =
1 2 3
4 5 6
7 8 9
A-1 = ?
A-1= Adj(A)
1
𝐴

16. Minor….?
 Minor of an element aij of matrix is the
determinant obtained by ignoring ith row and
jth column in which the element aij lies.
 Minor of an element aij is denoted by Mij .

17. How to find minor of matrix..?
There are three steps:-
1.Exclude the row and the column which contain the
particular element within the matrix.
2.Form a new smaller matrix with the remaining elements.
3.Find the determinant of the minor of each element of the
matrix.

18. Lets see how...
The minor of the element
a11 is as follows:
row
column
Similarly a23 = M23= ?

19. Cofactor….?

20. Adjoint… ?
 The adjoint of the square matrix is defined as the
transpose of the cofactor matrix of A.
 The adjoint of a matrix A is denoted by adjA.

21. Find the ad-joint of the following matrix
A=
1 2 3
0 2 4
0 0 5
SOLUTION:
1. Minor of 𝑀11 =
2 4
0 5
= (10-0) = 10
2. Minor of 𝑀12 =
0 4
0 5
= 0
3. Minor of 𝑀13 =
0 2
0 0
= 0
4. Minor of 𝑀21 =
2 3
0 5
= (10-3) = 7
5. Minor of 𝑀22 =
1 3
0 5
= (5-0) = 5
6. Minor of 𝑀23 =
1 2
0 0
= 0
7. Minor of 𝑀31 =
2 3
2 4
= (8-6) = 2
8. Minor of 𝑀32 =
1 3
0 4
= (4-0) = 4
9. Minor of 𝑀33 =
1 2
0 2
= (2-0) = 2
𝐴 =
1 2 3
0 2 4
0 0 5
= 1 10 − 0 − 2 0 + 3 = 10
𝐴−1 EXISTS

22. Cofactor of 𝑎11= ( −1)1 +1 10 = 10
Cofactor of 𝑎12= ( −1)1 +2
0 = 0
Cofactor of 𝑎13= ( −1)1+3
0 = 0
Cofactor of 𝑎21= ( −1)2 +1 2 = -7
Cofactor of 𝑎22= ( −1)2 +2
5 = 5
Cofactor of 𝑎23= ( −1)2+3
0 = 0
Cofactor of 𝑎31= ( −1)3+1
2 = 2
Cofactor of 𝑎32= ( −1)3 +2
4 = -4
Cofactor of 𝑎33= ( −1)3 +3
2= 2
Cofactor of matrix=
10 0 0
−10 5 0
2 −4 2
Adjoint of A=
10 −10 2
0 5 −4
0 0 2
𝐴−1 =
1
10
10 −10 2
0 5 −4
0 0 2
COFACTOR OF (𝑎𝑖𝑗) = ( −1)𝑖+𝑗
𝑀𝑖𝑗

23. APPLICATIONS
OF MATRICES

24. To solve linear equation
2 x - y = 4
3 x + 2 y= 13
Values of x and y matrices
2 −1
3 2
MATRICES FORM OF LINEAR EQUATION
𝑥
𝑦
4
13
4
13
4
13

25. 2 −1
3 2
𝑥
𝑦
4
13
2𝑥 − 𝑦
3𝑥 + 2𝑦
4
13
A X = B
𝐴−1 A X = 𝐴−1 B
I X = 𝐴−1 B
X = 𝐴−1
B

26. 𝐴 ≠ 0
|A| = (2 X 2) – (-1 X 3)
=4+3
=7
𝐴−1 exist

27. TO FIND 𝐴−1
2 −1
3 2
SOLUTION :-
1. MINOR OF THE ELEMENT (𝑀𝑖𝑗)
𝑀11 = 2
𝑀12= 3
𝑀21 = -1
𝑀22= 2
2. COFACTOR OF (𝑎𝑖𝑗) = ( −1)𝑖+𝑗
𝑀𝑖𝑗
Cofactor of 𝑎11= ( −1)1 +1 2 = 2
Cofactor of 𝑎12= ( −1)1 +2
3 = - 3
Cofactor of 𝑎21= ( −1)2+1
-1 = 1
Cofactor of 𝑎22= ( −1)2 +2 2 = 2
Cofactor matrix A =
2 −3
1 2
Adjoint A =
2 1
−3 2
𝐴−1 =
1
7
2 1
−3 2

28. X = 𝐴−1
B
𝑥
𝑦 =
1
7
2 1
−3 2
4
13
= 1
7
𝛿 + 3
−12 + 26
1
7
21
14
=
=
3
2
X = 3
Y = 2

29. THANK YOU