Determinants | Class 12 Mathematics Notes
By ConceptScroll Team · Published on 17 July 2026 · 4 min read
Determinants – this guide gives you a concise, exam-ready overview of Determinants from Class 12 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.
4.4 Minors and Cofactors
This section introduces the concepts of minors and cofactors, which are essential for expanding determinants of higher order matrices. The minor of an element a_ij in a determinant is defined as the determinant obtained by deleting the ith row and jth column containing that element, denoted by M_ij. For a determinant of order n (n ≥ 2), the minor is a determinant of order n - 1. The cofactor A_ij of element a_ij is defined as A_ij = (-1)^(i+j) × M_ij, incorporating a sign factor depending on the position of the element. Examples illustrate finding minors and cofactors for elements in 2×2 and 3×3 determinants. The section explains that the determinant of a matrix can be expanded as the sum of the products of elements of any row (or column) with their corresponding cofactors. It also notes that the sum of products of elements of one row (or column) with cofactors of a different row (or column) is zero, a property useful in proofs and computations.
📊 Diagram: No diagrams in this section.
🧪 Activity: No specific activity in this section.
🔗 Connection: Leads to the concept of adjoint and inverse matrices using cofactors.
Frequently asked questions
If the area of triangle is 4 square unit and vertices are (k, 0), (4, 0), (0, 2) then the value of k is..
0
If (6, 0), (4, 3), (2, 1) is the vertices of a triangle then the area of the triangle is
5 square unit
If any two rows or any two columns are identical or proportional, then the value of determinant is
0
Evaluate the determinants in Exercises 1 and 2. 1. \(\left| \begin{array}{cc} 2 & 4 \\ -5 & -1 \end{array} \right|\) 2. (i) \(\left| \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right|\) (ii) \(\left| \begin{array}{cc} x^{2} - x + 1 & x - 1 \\ x + 1 & x + 1 \end{array} \right|\) 3. If \(A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}\), then show that \(|2A| = 4 |A|\) 4. If \(A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}\), then show that \(|3A| = 27 |A|\) 5. Evaluate the determinants (i) \(\left| \begin{array}{ccc} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{array} \right|\) (ii) \(\left| \begin{array}{ccc} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{array} \right|\) (iii) \(\left| \begin{array}{lll}0 & 1 & 2\\ -1 & 0 & -3\\ -2 & 3 & 0 \end{array} \right|\) (iv) \(\left| \begin{array}{rrr}2 & -1 & -2\\ 0 & 2 & -1\\ 3 & -5 & 0 \end{array} \right|\) 6. If \(\mathrm{A} = \left[ \begin{array}{rrr}1 & 1 & -2\\ 2 & 1 & -3\\ 5 & 4 & -9 \end{array} \right]\) , find \(|\mathrm{A}|\) 7. Find values of \(x\) , if (i) \(\left| \begin{array}{lll}2 & 4\\ 5 & 1 \end{array} \right| = \left| \begin{array}{lll}2x & 4\\ 6 & x \end{array} \right|\) (ii) \(\left| \begin{array}{ll}2 & 3\\ 4 & 5 \end{array} \right| = \left| \begin{array}{ll}x & 3\\ 2x & 5 \end{array} \right|\) 8. If \(\left| \begin{array}{ll}x & 2\\ 18 & x \end{array} \right| = \left| \begin{array}{ll}6 & 2\\ 18 & 6 \end{array} \right|\) , then \(x\) is equal to (A) 6 (B) \(\pm 6\) (C) -6 (D) 0
1. Evaluate \(\left| \begin{array}{cc} 2 & 4 \\ -5 & -1 \end{array} \right| = (2)(-1) - (4)(-5) = -2 + 20 = 18.\n 2.(i) \(\left| \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right| = (\cos \theta)(\cos \theta) - (-\sin \theta)(\sin \theta) = \cos^2 \theta + \sin^2 \theta = 1.\n (ii) \(\left| \begin{array}{cc} x^{2} - x + 1 & x - 1 \\ x + 1 & x + 1 \end{array} \right| = (x^{2} - x + 1)(x + 1) - (x - 1)(x + 1) = (x^{3} + x^{2} - x^{2} - x + x + 1) - (x^{2}
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