Determinant of product is product of determinants
Dependencies:
- A matrix is full-rank iff its determinant is non-0
- Full-rank square matrix is invertible
- AB = I implies BA = I
- Full-rank square matrix in RREF is the identity matrix
- Elementary row operation is matrix pre-multiplication
- Matrix multiplication is associative
- Determinant of upper triangular matrix
Let
Proof
Case 1:
Assume
Therefore,
Case 2:
Therefore,
Dependency for:
Info:
- Depth: 11
- Number of transitive dependencies: 48
Transitive dependencies:
- /linear-algebra/vector-spaces/condition-for-subspace
- /linear-algebra/matrices/gauss-jordan-algo
- /sets-and-relations/equivalence-relation
- Group
- Ring
- Polynomial
- Integral Domain
- Comparing coefficients of a polynomial with disjoint variables
- 0x = 0 = x0
- Field
- Vector Space
- Linear independence
- Span
- A field is an integral domain
- Semiring
- Matrix
- Stacking
- System of linear equations
- Product of stacked matrices
- Matrix multiplication is associative
- Reduced Row Echelon Form (RREF)
- Submatrix
- Determinant
- Determinant of upper triangular matrix
- Swapping last 2 rows of a matrix negates its determinant
- Matrices over a field form a vector space
- Row space
- Elementary row operation
- Determinant after elementary row operation
- Every elementary row operation has a unique inverse
- Row equivalence of matrices
- Row equivalent matrices have the same row space
- RREF is unique
- Identity matrix
- Inverse of a matrix
- Inverse of product
- Elementary row operation is matrix pre-multiplication
- Row equivalence matrix
- Equations with row equivalent matrices have the same solution set
- Rank of a matrix
- Basis of a vector space
- Linearly independent set is not bigger than a span
- Homogeneous linear equations with more variables than equations
- Rank of a homogenous system of linear equations
- Full-rank square matrix in RREF is the identity matrix
- A matrix is full-rank iff its determinant is non-0
- Full-rank square matrix is invertible
- AB = I implies BA = I