Class 12 Maths Determinants – NCERT Formulas & Examples

Class 12 Maths Chapter 4: Determinants

Welcome to FUZY MATH ACADEMY. This chapter covers Determinants with all basic rules, formulas, examples, and 25 fully solved questions from NCERT to IIT level, along with 15 FAQs. Solutions are interactive and toggleable.

Basic Rules & Formulas

• Determinant of 2×2 matrix: |[[a,b],[c,d]]| = ad - bc
• Determinant of 3×3 matrix: |[[a,b,c],[d,e,f],[g,h,i]]| = a(ei-fh)-b(di-fg)+c(dh-eg)
• Interchanging two rows or columns changes the sign of determinant.
• Multiplying a row or column by k multiplies determinant by k.
• If a row or column is all zeros, determinant = 0.
• Determinant of triangular matrix = product of diagonal elements.
• Inverse using determinant: A^-1 = adj(A)/|A|, if |A| ≠ 0.

Examples

Example 1

Find |[[1,2],[3,4]]|

Solution: |A| = 1*4 - 2*3 = -2

Example 2

Find |[[1,2,3],[0,1,4],[5,6,0]]|

Solution: |A| = 1*(1*0-4*6) - 2*(0*0-4*5) + 3*(0*6-1*5) = 1

25 NCERT + IIT-Level Questions

Question 1

Find |[[2,3],[1,4]]|

|A| = 2*4 - 3*1 = 5

Question 2

Find |[[1,2,3],[4,5,6],[7,8,9]]|

|A|=0

Question 3

Find |[[1,0,2],[3,1,0],[2,3,1]]|

|A|=15

Question 4

Compute |[[2,1,3],[1,0,4],[5,2,1]]|

|A| = 9

Question 5

Find determinant of [[3,0,2],[2,0,-2],[0,1,1]]

|A| = 10

Question 6

If two rows of [[1,2,3],[4,5,6],[7,8,9]] are interchanged, find determinant.

|A| = -0 = 0

Question 7

Compute determinant of 3x3 upper triangular matrix [[1,2,3],[0,5,6],[0,0,9]]

|A| = 1*5*9 = 45

Question 8

Evaluate |[[1,1,1],[2,3,4],[1,0,1]]|

|A| = 2

Question 9

Check determinant of [[1,2,3],[0,1,4],[5,6,0]]

|A| = 1

Question 10

Compute determinant of [[2,1,0],[3,0,1],[1,4,2]]

|A| = 2*(0*2-1*4)-1*(3*2-1*1)+0*(3*4-0*1)=-8-5+0=-13

Question 11

Find |[[1,2,1],[0,1,0],[2,3,4]]|

|A| = 1*(1*4-0*3) - 2*(0*4-0*2) +1*(0*3-1*2) = 4-0-2=2

Question 12

Evaluate determinant of [[0,2,1],[1,3,0],[2,1,4]]

|A|=0*(3*4-0*1)-2*(1*4-0*2)+1*(1*1-3*2)=0-8-5=-13

Question 13

Find |[[1,0,2],[0,1,3],[4,5,6]]|

|A| = 1*(1*6-3*5)-0*(0*6-3*4)+2*(0*5-1*4)=1*(6-15)+0+2*(-4)=-9-8=-17

Question 14

Find determinant of [[2,1,3],[1,2,1],[3,0,2]]

|A| = 2*(2*2-1*0)-1*(1*2-1*3)+3*(1*0-2*3)=2*(4-0)-1*(2-3)+3*(0-6)=8-(-1)-18=-9

Question 15

Evaluate |[[1,2,3,4],[0,1,2,3],[1,0,1,2],[2,1,0,1]]|

Expansion using minors ⇒ |A|=-4

Question 16

Compute determinant of [[1,2,0],[3,0,1],[4,5,2]]

|A|=1*(0*2-1*5)-2*(3*2-1*4)+0*(3*5-0*4)=-5-2+0=-7

Question 17

Find determinant of [[2,3,1],[1,0,2],[3,4,5]]

|A|=2*(0*5-2*4)-3*(1*5-2*3)+1*(1*4-0*3)=-16-(-1)+4=-11

Question 18

Compute determinant of [[1,1,1],[1,2,3],[1,3,6]]

|A|=1*(2*6-3*3)-1*(1*6-3*1)+1*(1*3-2*1)=1*(12-9)-1*(6-3)+1*(3-2)=3-3+1=1

Question 19

Find |[[2,1,1],[1,3,2],[1,0,0]]|

|A|=2*(3*0-2*0)-1*(1*0-2*1)+1*(1*0-3*1)=0-(-2)-3=2-3=-1

Question 20

Compute determinant of [[1,2,3],[0,1,4],[5,6,0]]

|A| = 1*(1*0-4*6)-2*(0*0-4*5)+3*(0*6-1*5)= -24+40-15=1

Question 21

Evaluate determinant [[3,0,2],[2,0,-2],[0,1,1]]

|A|=10

Question 22

Compute |[[1,2,1],[2,3,4],[1,0,1]]|

|A|=1*(3*1-4*0)-2*(2*1-4*1)+1*(2*0-3*1)=3-(-2)-3=2

Question 23

Find determinant [[1,0,2],[0,1,3],[4,5,6]]

|A|=-17

Question 24

Compute |[[2,1,3],[1,2,1],[3,0,2]]|

|A|=-9

Question 25

Evaluate determinant [[1,1,1],[1,2,3],[1,3,6]]

|A|=1

15 FAQs

FAQ 1: What is a determinant?

A determinant is a scalar value associated with a square matrix, giving information about invertibility.

FAQ 2: How to calculate 2×2 determinant?

For [[a,b],[c,d]], |A| = ad - bc

FAQ 3: How to calculate 3×3 determinant?

Use Sarrus rule or cofactor expansion: |A| = a(ei-fh)-b(di-fg)+c(dh-eg)

FAQ 4: What happens if two rows are equal?

Determinant = 0

FAQ 5: What is a singular matrix?

A square matrix with determinant 0 is singular.

FAQ 6: What is a non-singular matrix?

A matrix with determinant ≠ 0 is non-singular and invertible.

FAQ 7: Can determinant be negative?