What is a determinant?

A determinant is a scalar value that can be computed from the elements of a square matrix and gives information about its invertibility and area/volume transformations.

How to find determinant of 3x3 matrix?

Use Sarrus Rule or expansion by minors to find determinant of 3×3 matrices.

Determinants Class 12 Maths Notes | Formulas, Theorems & Solved Examples

🔹 Introduction to Determinants

A Determinant is a scalar value that can be computed from a square matrix. It helps determine whether a matrix is invertible and plays a major role in coordinate geometry, vector transformations, and system of equations.

Representation of a 3×3 Determinant

🔹 Important Formulas

2×2 Determinant: |A| = ad − bc

3×3 Determinant: |A| = a₁₁(a₂₂a₃₃−a₂₃a₃₂) − a₁₂(a₂₁a₃₃−a₂₃a₃₁) + a₁₃(a₂₁a₃₂−a₂₂a₃₁)

Property 1: If any two rows or columns are identical, then |A| = 0

Property 2: Interchanging two rows changes the sign of determinant

Property 3: |Aᵀ| = |A|

Property 4: |AB| = |A| × |B|

🔹 Geometrical Meaning of Determinant

For a 2×2 matrix representing points in 2D, |A| gives the area of a parallelogram formed by column vectors.

Area of parallelogram = |Determinant|

🔹 Theorems on Determinants

Theorem 1: If a row (or column) of a determinant is multiplied by a constant k, then determinant becomes k times its original value.

Theorem 2: If any two rows (or columns) of a determinant are interchanged, its value changes sign.

Theorem 3: If all elements of a row are sum of two terms, determinant can be expressed as sum of two determinants.

🧮 Solved Examples (Step-by-Step)

Example 1: Find determinant of [[2,3],[1,4]]. → |A| = (2×4)−(3×1) = 8−3 = 5

Example 2: Find |A| for [[1,2,3],[2,4,6],[1,2,3]]. → Two rows are proportional ⇒ |A| = 0 ✅

Example 3: Find determinant of [[1,0,2],[−1,3,1],[3,2,4]]. → Expansion = 1(3×4−1×2) − 0 + 2(−1×2−3×3) = 10 − 26 = −16

Example 4: Show |Aᵀ| = |A| for [[1,2],[3,4]] ✅

Example 5: If A = [[1,2],[3,4]] and B = [[0,1],[2,3]], find |AB| = |A||B| = (−2)(−2) = 4

Example 6: Find area of triangle with vertices (1,2), (3,−1), (2,3). → Area = ½| x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂) | = ½|1(−1−3)+3(3−2)+2(2−(−1))| = ½|−4+3+6| = 2.5 sq. units

Example 7: Evaluate determinant by row transformation. [[1,2,1],[2,3,4],[3,4,5]] → R₂→R₂−2R₁, R₃→R₃−3R₁ → simplifies to upper triangular form → |A| = product of diagonal = 1×(−1)×(−1)=1

Example 8: If determinant is 0, show rows are linearly dependent ✅

Concept	Formula
2×2 Determinant	\|A\| = ad − bc
3×3 Determinant	Expansion by minors
Transpose	\|Aᵀ\| = \|A\|
Multiplication	\|AB\| = \|A\|\|B\|
Singular Matrix	\|A\| = 0

📘 Practice Questions

Find determinant of A = [[4,5],[2,3]]
Prove |Aᵀ| = |A|
Find area of triangle (1,2),(3,5),(5,1)
If |A|=2, find |3A|
Evaluate |[[2,3,1],[4,1,5],[1,2,3]]|

Answer Key

1️⃣ = 2
2️⃣ Verified
3️⃣ 6 sq. units
4️⃣ |3A| = 3²×|A| = 9×2 = 18
5️⃣ = −30

📗 Class 12 Maths – Determinants (Advanced Notes)