Determinants

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A determinant is a unique number (real or complex) associated with every square matrix. For a square matrix A, the determinant is denoted as |A|, det A, or Δ. Only square matrices have determinants.

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For a 1×1 matrix A = [a], the determinant is simply the element itself: |A| = a

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For a 2×2 matrix A = [[a₁₁, a₁₂], [a₂₁, a₂₂]], the determinant is: |A| = a₁₁a₂₂ - a₁₂a₂₁

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Using the formula |A| = a₁₁a₂₂ - a₁₂a₂₁: |A| = (3)(4) - (2)(1) = 12 - 2 = 10

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Expansion of a 3×3 determinant involves expressing it in terms of 2×2 determinants by multiplying each element of a chosen row/column by (-1)^(i+j) and its corresponding minor. All six possible expans

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|A| = a₁₁(-1)¹⁺¹|[a₂₂, a₂₃], [a₃₂, a₃₃]| + a₁₂(-1)¹⁺²|[a₂₁, a₂₃], [a₃₁, a₃₃]| + a₁₃(-1)¹⁺³|[a₂₁, a₂₂], [a₃₁, a₃₂]| = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂₁a₃₃ - a₂₃a₃₁) + a₁₃(a₂₁a₃₂ - a₂₂a₃₁)

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Expanding along first row: |A| = 2(4×0 - 1×2) - 1(0×0 - 1×1) + 3(0×2 - 4×1) = 2(-2) - 1(-1) + 3(-4) = -4 + 1 - 12 = -15

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The minor Mᵢⱼ of element aᵢⱼ is the determinant obtained by deleting the ith row and jth column from the original determinant.