Class 12 Mathematics Chapter 4: Determinants

CBSE 2026–27 | NCERT-aligned free resource

1. Determinant of a Square Matrix

A determinant assigns a scalar value to a square matrix. CBSE includes determinants up to order 3×3, along with minors, cofactors, adjoint, inverse and applications.

2. Minors and Cofactors

The minor Mᵢⱼ is obtained by deleting row i and column j. The cofactor is Cᵢⱼ=(−1)ⁱ⁺ʲMᵢⱼ.

3. Properties

• Interchanging two rows changes the sign of the determinant.
• If two rows are identical, determinant is zero.
• A common factor in a row can be taken outside.
• Adding a multiple of one row to another does not change the determinant.

4. Area of a Triangle

The determinant method gives the signed area from the coordinates of three vertices; the geometric area is the absolute value of half the determinant.

5. Adjoint and Inverse

For a nonsingular square matrix, A⁻¹ = adj(A)/|A|. Therefore |A| must be non-zero for the inverse to exist.

6. Linear Equations

Systems of two or three linear equations can be expressed as AX=B and solved using the inverse matrix when A is invertible.

Worked Example

For a 2×2 matrix with determinant ad−bc=5, the matrix is nonsingular and an inverse exists. If the determinant is zero, the inverse does not exist.

Practice

1. Evaluate determinants using properties.
2. Find minors and cofactors.
3. Calculate the area of a triangle from coordinates.
4. Find adjoint and inverse of a 2×2 matrix.
5. Solve a system of linear equations using matrix inverse.

Deep-Dive Revision & Worked Examples

Determinant of a 2×2 Matrix

For A=[[a,b],[c,d]], |A|=ad−bc. If |A|=0, the matrix is singular and has no inverse.

Worked Example

For A=[[2,3],[1,4]], |A|=8−3=5. Since the determinant is non-zero, A is invertible.

Area Interpretation

The absolute value of the determinant formed by two planar vectors gives the area of the parallelogram generated by those vectors. Half of it gives the triangle area.

Geometric Visual

Solving Linear Equations

Determinants can be used through Cramer’s rule when the determinant of the coefficient matrix is non-zero. Always check consistency and the value of the determinant before applying the rule.

Practice

1. Evaluate determinants by expansion.
2. Use determinant properties to simplify calculations.
3. Find the area of a triangle from coordinates.
4. Solve a three-variable system using determinants.
5. Determine whether a matrix is singular.

Common Mistakes — Chapter 4

• Changing the sign incorrectly during cofactor expansion.
• Confusing determinant value with the determinant matrix.
• Using Cramer’s rule when the coefficient determinant is zero.