Determinants

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A determinant of order 2 is a square arrangement of 4 elements in 2 rows and 2 columns. General form: |a b| |c d| Value = ad - bc Example: |3 5| = (3)(2) - (5)(1) = 6 - 5 = 1

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General form: |a₁₁ a₁₂ a₁₃| ← R₁ |a₂₁ a₂₂ a₂₃| ← R₂ |a₃₁ a₃₂ a₃₃| ← R₃ ↑ ↑ ↑ C₁ C₂ C₃ where aᵢⱼ represents the element in ith row and jth column. Example: a₃₁ is the element in 3rd row, 1st

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Step 1: Use first row expansion formula: D = a₁₁|a₂₂ a₂₃| - a₁₂|a₂₁ a₂₃| + a₁₃|a₂₁ a₂₂| |a₃₂ a₃₃| |a₃₁ a₃₃| |a₃₁ a₃₂| Step 2: Substitute values: D = 3|2 1| - (-2)|1 1| + 4|1 2|

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Triangle Property: If all elements of a determinant above OR below the main diagonal are zero, then the value equals the product of diagonal elements. Example 1 (Upper Triangle): |3 0 0| |5 4 0| = 3

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Property: When two rows (or columns) are interchanged, the determinant changes sign only. Example: A = |1 2 3| = -17 |4 -1 2| |0 2 -1| After R₁ ↔ R₃: A₁ = |0 2 -1| = 17 = -A |4 -

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Property: If any two rows (or columns) are identical, the determinant value is zero. Example: |1 2 3| |1 2 3| = 0 (R₁ and R₂ are identical) |4 5 6| Reason: If we interchange identical rows, t

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Property: If each element of one row (or column) is multiplied by constant k, the new determinant = k × original determinant. Example: A = |1 2 3| = 5 |0 1 2| |1 0 1| Multiply R₁ by 3

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Property: Adding k times any row to another row doesn't change the determinant value. Example: A = |1 2 3| = -8 |1 -2 0| |1 2 1| Perform R₁ → R₁ + 2R₂: A₁ = |3 -2 3| = -8 (sa