Determinants Solved Examples (Class 12 Mathematics)
Determinants numericals cover properties of determinants, expansion methods, and applications. These problems develop skills for solving linear systems usi
TL;DR: Determinants numericals cover properties of determinants, expansion methods, and applications. These problems develop skills for solving linear system…
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Determinants numericals cover properties of determinants, expansion methods, and applications. These problems develop skills for solving linear systems usi
Determinants — Solved Numerical Examples (Step by Step)
Example 1: Find the determinant of [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
Solution: Expanding along first row: det = 1×det([[5, 6], [8, 9]]) - 2×det([[4, 6], [7, 9]]) + 3×det([[4, 5], [7, 8]]). = 1×(45-48) - 2×(36-42) + 3×(32-35) = 1×(-3) - 2×(-6) + 3×(-3) = -3 + 12 - 9 = 0.
Example 2: Using Cramer's rule, solve: 2x + y = 5 and x + 3y = 6.
Solution: Coefficient matrix A = [[2, 1], [1, 3]]. det(A) = 6 - 1 = 5. For x: det([[5, 1], [6, 3]]) = 15 - 6 = 9. x = 9/5. For y: det([[2, 5], [1, 6]]) = 12 - 5 = 7. y = 7/5.
Example 3: Verify if the system 2x + 3y = 5, 4x + 6y = 10 has a unique solution.
Solution: Coefficient matrix [[2, 3], [4, 6]]. det = 2×6 - 3×4 = 12 - 12 = 0. Since det = 0, the matrix is singular and the system does not have a unique solution (either no solution or infinite solutions). Checking: second equation is 2 times the first, so they are the same line. Infinite solutions.
Example 4: Find the area of triangle with vertices (1, 2), (3, 4), (5, 2).
Solution: Area = 0.5 × |det([[1, 2, 1], [3, 4, 1], [5, 2, 1]])| = 0.5 × |det([[1, 2], [3, 4]] - 3×det([[2, 1], [2, 1]]) + 5×det([[2, 1], [4, 1]])|. Using first vertex approach: Area = 0.5 × |1(4-2) - 2(3-5) + 1(6-20)| = 0.5 × |2 + 4 - 14| = 0.5 × |-8| = 4.
Example 5: If det(A) = 5, find det(3A) for a 2×2 matrix A.
Solution: For an n×n matrix, det(kA) = k^n × det(A). For 2×2 matrix: det(3A) = 3² × det(A) = 9 × 5 = 45.
Tips
- Determinant properties: det(A × B) = det(A) × det(B), det(kA) = k^n × det(A) for n×n matrix.
- If det = 0, matrix is singular and not invertible.
- Cramer's rule: for Ax = b, xi = det(Ai) / det(A) where Ai is A with ith column replaced by b.
Frequently Asked Questions
When should we use Cramer's rule?
Cramer's rule is useful for solving small systems (2×2 or 3×3) with a unique solution (det ≠ 0). For larger systems, Gaussian elimination is more efficient.
What does det(A) = 0 mean geometrically?
det(A) = 0 means the matrix represents a singular transformation that collapses the space to a lower dimension. Rows/columns are linearly dependent.
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