# Determinants Notes, Class 12 Mathematics | SwaVid Determinants, NCERT Class 12 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers. Canonical: https://www.swavid.com/cbse/class-12/maths/chapter/determinants Source: https://www.swavid.com/cbse/class-12/maths/chapter/determinants # Determinants ## Formulas and definitions ## Key ideas ## Common mistakes ## Quick checks ## Keep going ## Questions parents ask ## Class 12 Mathematics on SwaVid ### Determinant of a square matrix ### Area of a triangle ### Minors and cofactors ### Adjoint and inverse of a matrix ### Solving linear systems ### Next chapter ### Related chapter ### Related chapter ### All chapters SwaVid › CBSE › Class 12 › Mathematics › Determinants Notes written by SwaVid from the NCERT textbook. Class 12 Mathematics, Chapter 4. Key ideas, formulas and common mistakes. CBSE does not print marks for a single chapter. Short answer. This chapter defines the determinant of a square matrix and covers its evaluation up to order three along with minors and cofactors. It uses determinants to calculate the area of a triangle, find the adjoint and inverse of a matrix, and solve systems of linear equations using the matrix method. Source: NCERT Mathematics Part I (Reprint 2026-27) , Chapter 4, Determinants A determinant is a unique number associated with any square matrix. For a 2 × 2 matrix, it is computed by cross-multiplying diagonal entries, and for a 3 × 3 matrix, by expanding along any row or column. The area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is evaluated using a 3 × 3 determinant. Since area is a positive quantity, the absolute value of the determinant expression is taken, and collinear points yield an area of zero. The minor Mᵢⱼ of an element is the determinant obtained by deleting its i-th row and j-th column. The cofactor Aᵢⱼ is given by (-1)ⁱ⁺ʲ Mᵢⱼ, and expanding along any row or column is equivalent to summing each element multiplied by its cofactor. The adjoint of a square matrix A is the transpose of its matrix of cofactors. If the determinant |A| is non-zero, the matrix is non-singular and its inverse exists, given by (1 / |A|) adj A. A system of linear equations AX = B can be solved using the matrix method. When |A| ≠ 0, the system is consistent and has a unique solution X = A⁻¹ B. Try each question first. Then tap it to see the answer. |A| = (2)(2) - (4)(-1) |A| = 4 - (-4) |A| = 4 + 4 = 8 8 Area = (1/2) |3(2 - 1) + (-4)(1 - 8) + 5(8 - 2)| Area = (1/2) |3(1) - 4(-7) + 5(6)| Area = (1/2) |3 + 28 + 30| Area = (1/2) |61| = 61/2 61/2 sq units Use the formula |adj A| = |A|ⁿ⁻¹ where n = 3 |adj A| = 5³⁻¹ |adj A| = 5² = 25 25 Continuity and Differentiability . Matrices . Linear Programming . The 13 chapters of Class 12 Mathematics . CBSE does not give marks to one chapter. It prints marks for units or parts, and the chapter list page shows them all. No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed. Continuity and Differentiability. SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works. - Ch 4 NCERT book - 80 marks, theory paper - 3 quick checks - Writing |kA| = k|A| for a matrix of order n . Factoring a scalar k out of a determinant pulls k from every row, so |kA| = kⁿ |A| for an n × n matrix. - Forgetting to transpose the cofactor matrix to get the adjoint . Calculate all cofactors Aᵢⱼ, arrange them into a matrix, and then take the transpose to obtain adj A. - Omitting the absolute value sign when calculating triangle area . Determinant values can be negative, but area must be positive, so take the absolute value of the final numerical expression. - Attempting to invert a singular matrix . Always check that |A| ≠ 0 before writing A⁻¹ = (1 / |A|) adj A. If |A| = 0, the inverse does not exist. ## Key Links - [CBSE](https://www.swavid.com/cbse) - [Class 12](https://www.swavid.com/cbse/class-12) - [Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup) - [All 13 chapters](https://www.swavid.com/cbse/class-12/maths/chapters) - [NCERT Mathematics Part I (Reprint 2026-27)](https://ncert.nic.in/textbook.php) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/continuity-and-differentiability) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/matrices) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/linear-programming) - [Class 12 Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup)