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.
4.6 Applications of Determinants and Matrices
This section focuses on applying determinants and matrices to solve systems of linear equations and analyze their consistency. A system is consistent if it has at least one solution; inconsistent if no solution exists. The section discusses solving linear systems using the inverse of the coefficient matrix. Given a system of three equations in three variables, expressed as A X = B, if A is nonsingular (|A| ≠ 0), then the unique solution is X = A^{-1} B. If A is singular (|A| = 0), the system may be inconsistent or have infinitely many solutions, depending on whether (adj A) B is zero or not. Examples demonstrate solving 2×2 and 3×3 systems using matrix inverses, including calculation of determinants, adjoints, and inverses, followed by matrix multiplication to find solutions. The section also includes a real-world problem involving costs of items solved by matrix methods. Miscellaneous examples illustrate matrix multiplication and inverse properties. Exercises encourage practice in checking consistency and solving systems using determinants and matrix inverses.
📊 Diagram: No diagrams in this section.
🧪 Activity: No specific activity in this section.
🔗 Connection: Leads to exercises and miscellaneous problems reinforcing determinant applications.
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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