Evaluating 2×2 and 3×3 determinants: Class 12 practice

What you should know first

• Integer arithmetic
• Matrix notation

The idea behind evaluating 2×2 and 3×3 determinants

|a b; c d| = ad − bc. A 3×3 determinant is expanded along a row using cofactors with the sign pattern + − +. A zero determinant means the matrix has no inverse.

Step-by-step method

1. 2×2: ad − bc.
2. 3×3: expand along row 1: a(ei − fh) − b(di − fg) + c(dh − eg).
3. Choose a row with zeros to save work.

Worked example 1 (Basic)

Evaluate |-2 -1; -4 -2|.

1. = (-2)(-2) − (-1)(-4)
2. = 0
3. Answer: 0

Worked example 2 (Standard)

Evaluate the determinant |-2 2 1; -2 3 -1; -4 4 -2|.

1. = -2(-2) − 2(0) + 1(4)
2. = 8
3. Answer: 8

Worked example 3 (Challenge)

Evaluate the determinant |3 -3 0; -4 4 3; 4 4 5|.

1. = 3(8) − -3(-32) + 0(-32)
2. = -72
3. Answer: -72

Common mistakes and misconceptions

• Forgetting the minus sign on the middle cofactor
• Computing ad + bc

Practice questions

1. (Basic) Evaluate |1 5; 0 -1|.
2. (Basic) Evaluate |1 1; -2 2|.
3. (Standard) Evaluate the determinant |-3 -1 0; -3 -3 5; 0 0 0|.
4. (Standard) Evaluate the determinant |-3 -2 -2; 0 -4 4; 4 2 -4|.
5. (Challenge) Evaluate the determinant |5 3 -4; 0 5 -4; 4 0 -3|.
6. (Challenge) Evaluate the determinant |-4 1 1; 0 4 3; -1 2 4|.

Answer key with full working

• Q1: = (1)(-1) − (5)(0) → = -1 → Answer: -1
• Q2: = (1)(2) − (1)(-2) → = 4 → Answer: 4
• Q3: = -3(0) − -1(0) + 0(0) → = 0 → Answer: 0
• Q4: = -3(8) − -2(-16) + -2(16) → = -88 → Answer: -88
• Q5: = 5(-15) − 3(16) + -4(-20) → = -43 → Answer: -43
• Q6: = -4(10) − 1(3) + 1(4) → = -39 → Answer: -39

Syllabus note

This skill is listed under "Determinants" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

• Determinants chapter notes (Class 12)
• Inverse of a 2×2 matrix
• Solving equations with matrices / Cramer's rule
• Derivatives of polynomials and slope of a tangent
• Definite integrals of polynomials
• Matrix multiplication
• Conditional probability and Bayes' theorem
• All Class 12 practice

Frequently asked questions

Which chapter is evaluating 2×2 and 3×3 determinants in?

"Determinants", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.