Class 12th determinants notes for jee mains and advance

Revision Notes for Class 12 Mathematics

Chapter 4 – Determinants


Matrix Representation of Linear Equations

When a system of algebraic equations is given to us as:

a1 x + b1 y = c1

a2 x + b2 y = c2

Then we can express them in the form of matrices as:

 a1 b1   x   c1 
a =
 2 b2   y  c2 

To get the solution of a system of linear equations, we find all the values of the variables
satisfying all the linear equations in the system.



Definition of Determinants

• We can define the determinant of a matrix as a scalar value that can be calculated
from the elements of a square matrix.

a b 
• The scalar value for a square matrix  1 1  is given by a1b2 − a2b1 .
 a2 b2 

• It is represented as A or det ( A ) or  .




www.vedantu.com 1

, a b  a b
• For a matrix  1 1  , the determinant is written as 1 1 .
 a2 b2  a2 b2

• Square matrices are those matrices that have the same number of rows and columns.
Only such matrices have determinants.



Types of Determinants

1. First Order Determinant – It is the determinant of a matrix of order one. The element of
the matrix will be the determinant value.

For example,

 2  2  2

2. Second Order Determinant - It is the determinant of a matrix of order two.

a b  a b
If  1 1  , then 1 1 = a1b2 − a2b1 .
 a2 b2  a2 b2

For example,

1 3  1 3
5 3   5 3
 

 (1 3) − ( 3  5 )  3 − 15  −12

3. Third Order Determinant - It is the determinant of a matrix of order three.

 a1 b1 c1 
Let us consider  a2 b2 c2  .
 a3 b3 c3 




www.vedantu.com 2