📘 Chapter: Determinants (Class 12 – NCERT Maths)
🔹 1. Introduction
Determinants are numerical values associated with square matrices only. They help us understand important properties of matrices such as:
✔ Whether a matrix has an inverse
✔ Whether a system of linear equations has a unique solution
✔ Geometrical results like area of a triangle
A determinant is denoted by |A| or det(A).
🔹 2. Determinant of a Square Matrix
🧮 (a) Determinant of a 2 × 2 Matrix
For a matrix:
A = [ a b
c d ]
The determinant is:
|A| = ad − bc
📌 If ad − bc = 0, the matrix is singular.
🧮 (b) Determinant of a 3 × 3 Matrix
For a matrix:
A = [ a b c
d e f
g h i ]
Using expansion by minors:
|A| = a(ei − fh) − b(di − fg) + c(dh − eg)
✨ Expansion can be done along any row or column, whichever is easier.
🔹 3. Minors and Cofactors
🔸 Minor (Mij): The determinant obtained by deleting the i-th row and j-th column of the given matrix.
🔸 Cofactor (Aij):
Aij = (−1)i+j Mij
📌 Cofactors are essential for:
✔ Finding adjoint of a matrix
✔ Expanding determinants
🔹 4. Properties of Determinants
📌 Important properties to remember for exams:
1️⃣ Interchanging two rows or columns changes the sign of the determinant.
2️⃣ If any two rows or columns are identical, the determinant is zero.
3️⃣ If a row or column is multiplied by a constant k, the determinant is also multiplied by k.
4️⃣ If one row or column is the sum of two rows or columns, the determinant can be split into two determinants.
5️⃣ If a matrix contains a row or column of all zeros, its determinant is zero.
6️⃣ Adding a multiple of one row to another row does not change the determinant.
🔹 5. Area of a Triangle Using Determinants 📐
The area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is:
Area = 1/2 | x₁ y₁ 1
x₂ y₂ 1
x₃ y₃ 1 |
📌 If the value of this determinant is zero, the points are collinear.
🔹 6. Adjoint of a Matrix 🔁
The adjoint of a matrix is the transpose of its cofactor matrix.
🪜 Steps to find adjoint:
1️⃣ Find minors of all elements.
2️⃣ Convert minors into cofactors.
3️⃣ Arrange cofactors in matrix form.
4️⃣ Take the transpose of the matrix.
🔹 7. Inverse of a Matrix Using Determinants
The inverse of a square matrix A exists only if:
|A| ≠ 0
Formula for inverse:
A−1 = (1 / |A|) × adj A
🚫 If |A| = 0, the matrix is singular and has no inverse.
🔹 8. Applications of Determinants 💡
Determinants are used to:
✔ Find inverse of matrices
✔ Solve systems of linear equations
✔ Check consistency of equations
✔ Find area of geometric figures
🔹 9. Solving Linear Equations Using Inverse Method
A system of linear equations can be written as:
AX = B
If |A| ≠ 0, then:
X = A−1 B
📌 This method works only when the coefficient matrix is square and invertible.
⭐ Key Takeaways
✔ Determinants are defined only for square matrices.
✔ Zero determinant implies no inverse.
✔ Properties of determinants help simplify calculations.
✔ Very important chapter for board exams and entrance tests.