Class 12 Maths Chapter 4 Determinants

Determinants are closely connected to matrices, which you studied in the previous chapter. If matrices are a way to represent numbers in rows and columns, then determinants are the special values derived from those matrices. 

They give us essential information about the properties of a matrix and are essential tools for solving systems of linear equations, finding inverses and understanding geometric transformations.

Determinants Notes Class 12 PDF

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Definition and Basics

A determinant is just a single number (scalar value) calculated from a square matrix.

• It is written using vertical bars | |
• If the matrix is A, its determinant is written as |A| or det(A)
• Only square matrices (same number of rows and columns) can have determinants

Determinant of a 2×2 Matrix                            

If A = [a b; c d], the determinant is found using: |A| = (a × d) − (b × c)                                                

Example: If A = [3 2; 1 4], then: |A| = (3×4) − (2×1) = 12 − 2 = 10. This number, 10, is the determinant of the matrix.

Determinant of a 3×3 Matrix                                                                                                               

If A = [a b c; d e f; g h i], then the determinant is: |A| = a(ei − fh) − b(di − fg) + c(dh − eg) 

Example: A = [2 1 3; 4 0 5; 1 2 2]
|A| = 2(0×2 − 5×2) − 1(4×2 − 5×1) + 3(4×2 − 0×1)
= 2(−10) − 1(8 − 5) + 3(8 − 0)
= −20 − 3 + 24 = 1.                                                                                                                       

So, |A| = 1.                                                                                                                                                   

A determinant can be positive, negative, or zero. It all depends on the numbers and their positions.

Properties of Determinants

These properties make calculations much easier, so remember them well.

• Swapping any two rows or columns changes the sign of the determinant
• If two rows or columns are identical, the determinant is zero
• Multiplying all elements of a row or column by k multiplies the determinant by k
• If one row or column is a multiple of another, the determinant is zero
• Adding a multiple of one row or column to another does not change the determinant
• The determinant of a matrix is equal to that of its transpose
  
• |A| = |Aᵀ|

Expansion of Determinants

For 3×3 or bigger matrices, we use minors and cofactors to find determinants.

• A minor is the determinant of the smaller matrix formed by deleting one row and one column
• A cofactor is the minor multiplied by (−1)^(row + column)

Example:
Find the determinant of A = [2 3 1; 4 1 5; 3 2 2].

|A| = 2[(1×2 − 5×2)] − 3[(4×2 − 5×3)] + 1[(4×2 − 1×3)]
= 2(2 − 10) − 3(8 − 15) + 1(8 − 3)
= 2(−8) − 3(−7) + 1(5)
= −16 + 21 + 5 = 10.

So, |A| = 10.

This method works for any size of square matrix, though for higher-order matrices, the process becomes lengthier.

Application of Determinants

Determinants are not just abstract mathematical ideas, they have real applications:

1. To check if a matrix has an inverse: If |A| ≠ 0, then A⁻¹ exists; if |A| = 0, no inverse exists.
2. To solve linear equations using Cramer’s Rule: Determinants simplify equations into numerical form.
3. To find area and volume: In coordinate geometry, determinants help find the area of a triangle or the volume of a parallelepiped.
4. To check collinearity: If three points lie on the same straight line, the determinant formed by their coordinates equals zero.‍
5. In transformations: Determinants help understand how shapes change when they are rotated, stretched or reflected.

Cramer’s Rule

Cramer’s Rule is an application of determinants used to solve systems of linear equations. It provides an easy method to find the values of variables.

For example, for the system: a₁x + b₁y = c₁
a₂x + b₂y = c₂
We form determinants as follows:

D = |a₁ b₁; a₂ b₂|
Dₓ = |c₁ b₁; c₂ b₂|
Dᵧ = |a₁ c₁; a₂ c₂|

Then,

x = Dₓ / D and y = Dᵧ / D, provided D ≠ 0.

Example: 2x + 3y = 8 and 3x + 4y = 11

D = |2 3; 3 4| = (2×4 − 3×3) = −1
Dₓ = |8 3; 11 4| = (8×4 − 3×11) = 32 − 33 = −1
Dᵧ = |2 8; 3 11| = (2×11 − 8×3) = 22 − 24 = −2
So, x = Dₓ/D = (−1)/(−1) = 1 and y = Dᵧ/D = (−2)/(−1) = 2.

Hence, x = 1 and y = 2.

Determinants may seem complex initially, but with practice, they become simple and intuitive. They form the foundation for many important mathematical topics such as calculus, geometry and linear algebra. 

Once you grasp their meaning and logic, you will be able to solve questions quickly and confidently. Understanding determinants builds analytical and logical skills that go far beyond school mathematics.

Frequently Asked Questions

Q1. What is a determinant?

Ans. A determinant is a special number obtained from a square matrix that provides important information about the matrix, such as whether it has an inverse and if equations have unique solutions.

Q2. What does it mean if a determinant is zero?

Ans. It means that the matrix is singular and does not have an inverse. For systems of equations, it indicates that there may be no solution or infinitely many solutions.

Q3. How do minors and cofactors help in finding determinants?

Ans. Minors and cofactors help expand determinants of higher-order matrices. A minor is the determinant of a smaller matrix formed by deleting one row and column, while a cofactor adds a sign pattern to the minor.

Q4. What is the geometric meaning of a determinant?

Ans. In geometry, the determinant represents area or volume. For instance, the determinant of points in a plane can give the area of a triangle.

Q5. What is Cramer’s Rule?

Ans. Cramer’s Rule uses determinants to find the solution of linear equations by dividing specific determinants derived from coefficients and constants.