Class 12 Maths Determinants Notes

4.1 Introduction

Determinants are special numbers associated with square matrices. They help in finding the inverse of a matrix, solving linear equations, and calculating the area of a triangle using coordinates. Determinants play an important role in algebra and higher mathematics.

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4.2 Determinant

The determinant of a square matrix A is denoted by |A|.

🔹 Determinant of a 2 × 2 Matrix

If$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$A=[ac​bd​]

Then,$$|A| = ad – bc$$∣A∣=ad−bc

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🔹 Determinant of a 3 × 3 Matrix

For a 3 × 3 matrix, the determinant is calculated using expansion along a row or column.

🔹 Properties of Determinants

• Determinant of the identity matrix = 1
• If two rows or columns are equal, the determinant is zero
• Interchanging two rows or columns changes the sign of the determinant
• If one row is a multiple of another row, determinant is zero

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4.3 Area of a Triangle

The area of a triangle formed by three points
$(x_1, y_1), (x_2, y_2), (x_3, y_3)$(x1​,y1​),(x2​,y2​),(x3​,y3​)
is given by:$$\text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|$$Area=21​​​x1​x2​x3​​y1​y2​y3​​111​​​

🔹 Collinearity of Points

• If the area is zero, the three points lie on the same straight line.

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4.4 Minors and Cofactors

🔹 Minor

The minor of an element of a determinant is obtained by deleting the row and column containing that element.

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🔹 Cofactor

The cofactor of an element is given by:$$C_{ij} = (-1)^{i+j} M_{ij}$$Cij​=(−1)i+jMij​

where $M_{ij}$Mij​ is the minor of the element.

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4.5 Adjoint and Inverse of a Matrix

🔹 Adjoint of a Matrix

The adjoint of a square matrix A is the transpose of its cofactor matrix and is denoted by Adj(A).

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🔹 Inverse of a Matrix

If |A| ≠ 0, then the inverse of matrix A is:$$A^{-1} = \frac{1}{|A|} \, \text{Adj}(A)$$A−1=∣A∣1​Adj(A)

• Inverse exists only for non-singular matrices
• If |A| = 0, the matrix has no inverse

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4.6 Applications of Determinants and Matrices

🔹 1. Solving System of Linear Equations

Using Cramer’s Rule, determinants help solve simultaneous linear equations.

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🔹 2. Finding Inverse of a Matrix

Determinants are used to calculate the inverse of matrices, which is useful in many mathematical problems.

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🔹 3. Geometrical Applications

• Finding the area of a triangle
• Checking whether points are collinear

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✅ Key Points to Remember

Determinants simplify many algebraic and geometrical problems

Determinants are defined only for square matrices

A matrix is invertible only if its determinant is not zero

Cramer’s Rule is applicable only when |A| ≠ 0