Determinants
Odisha Board · Class 12 · Mathematics
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Calculate the determinant of the matrix A = [[3, 5], [1, 2]]
Find the determinant of the matrix B = [[4, 0], [7, -2]]
Calculate the determinant of the 3×3 matrix: [[1, 2, 3], [0, 4, 5], [0, 0, 6]]
Find the area of triangle with vertices A(1, 2), B(3, 4), and C(5, 1) using determinants.
Sample Questions
If det(A) = 6 for a 2×2 matrix A, what is det(3A)?
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54
For an n×n matrix A, if we multiply by scalar k, then det(kA) = k^n × det(A). Here n=2, so det(3A) = 3² × 6 = 9 × 6 = 54. Common mistake: Calculating 3 × 6 = 18 instead of 3² × 6.
Which of the following properties are true for determinants? (Select all correct answers)
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det(AB) = det(A) × det(B), det(A^T) = det(A), If two rows are identical, det(A) = 0
Correct properties: det(AB) = det(A)det(B), det(A^T) = det(A), and identical rows make det = 0. Wrong: det(A+B) ≠ det(A)+det(B) and det(kA) = k^n×det(A), not k×det(A).
Calculate the minor M₁₂ of the matrix [[2, 3, 1], [4, 5, 6], [7, 8, 9]]
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-6
Minor M₁₂ is obtained by deleting row 1 and column 2, giving matrix [[4, 6], [7, 9]]. M₁₂ = det([[4, 6], [7, 9]]) = 4×9 - 6×7 = 36 - 42 = -6.
If the cofactor A₁₁ = 8 for element a₁₁ in a 3×3 matrix, what is the minor M₁₁?
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8
Cofactor A₁₁ = (-1)^(1+1) × M₁₁ = (-1)² × M₁₁ = 1 × M₁₁ = M₁₁. Since A₁₁ = 8, therefore M₁₁ = 8. The sign factor (-1)^(i+j) for position (1,1) is positive.
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