Properties of Determinants in Linear Algebra

Properties of Determinants

Multiplicativity

• The determinant of the product of two matrices is the product of their determinants: det(AB) = det(A) * det(B)
• This property can be extended to the product of multiple matrices

Linearity in Rows/Columns

• If two rows/columns of a matrix are interchanged, the determinant changes sign
• If a row/column of a matrix is multiplied by a scalar, the determinant is also multiplied by that scalar
• If a row/column of a matrix is added to another row/column, the determinant does not change

Homogeneity

• If all elements of a row/column of a matrix are multiplied by a scalar, the determinant is also multiplied by that scalar

Invariance Under Elementary Operations

• The determinant does not change under elementary row operations (swap, multiply, add)
• The determinant does not change under elementary column operations (swap, multiply, add)

Zero Determinant

• If a matrix has a zero row or column, its determinant is zero
• If a matrix has a linearly dependent row or column, its determinant is zero

Determinant of Inverse

• The determinant of the inverse of a matrix is the reciprocal of its determinant: det(A^(-1)) = 1 / det(A)

Determinant of Transpose

• The determinant of the transpose of a matrix is the same as the determinant of the original matrix: det(A^T) = det(A)

Properties of Determinants

Multiplicativity

• Determinant of a product of two matrices is equal to the product of their individual determinants
• This property can be extended to the product of multiple matrices

Linearity in Rows/Columns

• Interchanging two rows or columns of a matrix results in a change of sign of the determinant
• Multiplying a row or column by a scalar results in the determinant being multiplied by that scalar
• Adding a row or column to another row or column does not change the determinant

Homogeneity

• Multiplying all elements of a row or column by a scalar results in the determinant being multiplied by that scalar

Invariance Under Elementary Operations

• Elementary row operations (swap, multiply, add) do not change the determinant
• Elementary column operations (swap, multiply, add) do not change the determinant

Zero Determinant

• A matrix with a zero row or column has a determinant of zero
• A matrix with linearly dependent rows or columns has a determinant of zero

Determinant of Inverse and Transpose

• The determinant of the inverse of a matrix is the reciprocal of its determinant
• The determinant of the transpose of a matrix is equal to the determinant of the original matrix

This quiz covers the properties of determinants, including multiplicativity and linearity in rows and columns. Learn how to calculate determinants and understand their applications in linear algebra.