Determinant Properties in Linear Algebra

Determinant Properties

Multiplicative Property

• det(AB) = det(A)det(B) for square matrices A and B
• This property allows us to break down the calculation of the determinant of a matrix product into the product of the determinants of the individual matrices.

Additive Property in Rows/Columns

• det(A + B) ≠ det(A) + det(B) in general
• However, if B is obtained by adding a multiple of one row/column of A to another row/column, then det(A + B) = det(A)
• This property is useful for finding the determinant of a matrix by using row/column operations.

Scalar Multiplication Property

• det(kA) = k^n det(A) for a square matrix A and scalar k, where n is the size of the matrix
• This property shows that the determinant of a matrix is scaled by a factor of k^n when the matrix is multiplied by a scalar k.

Invariance under Elementary Row/Column Operations

• The determinant remains unchanged under elementary row/column operations:
  • Swap two rows/columns: det(A) = det(A')
  • Multiply a row/column by a non-zero scalar: det(A) = det(A')
  • Add a multiple of one row/column to another row/column: det(A) = det(A')

Invariance under Matrix Equivalence

• The determinant remains unchanged under matrix equivalence:
  • det(PAQ) = det(A) for invertible matrices P and Q
  • This property shows that the determinant is a property of the matrix itself, not just its representation.

Multilinearity

• The determinant is a multilinear function, meaning it is linear in each row/column separately:
  • det(A + B, C, ..., Z) = det(A, C, ..., Z) + det(B, C, ..., Z)
  • det(kA, C, ..., Z) = k det(A, C, ..., Z)