Determinants — Chapter Summary

A determinant is a real

A determinant is a real or complex number associated with every square matrix A, denoted as |A|, det(A), or Δ. It is a function that maps square matri

For a matrix A = [a]

For a matrix A = [a] of order 1×1, the determinant is simply |A| = a. This is the simplest case where the determinant equals the single element of the

For a 2×2 matrix [[a₁₁

For a 2×2 matrix [[a₁₁, a₁₂], [a₂₁, a₂₂]], the determinant is |A| = a₁₁a₂₂ - a₁₂a₂₁. This represents the product of main diagonal elements minus the p

For a 3×3 matrix

For a 3×3 matrix, the determinant can be calculated by expanding along any row or column. When expanding along row 1: |A| = a₁₁(a₂₂a₃₃ - a₃₂a₂₃) - a₁₂

The minor M_ij of element a_ij

The minor M_ij of element a_ij is the determinant of the matrix obtained by deleting the i-th row and j-th column where a_ij is located. For a 3×3 mat