Determinant of product is product of determinants

Dependencies:

  1. A matrix is full-rank iff its determinant is non-0
  2. Full-rank square matrix is invertible
  3. AB = I implies BA = I
  4. Full-rank square matrix in RREF is the identity matrix
  5. Elementary row operation is matrix pre-multiplication
  6. Matrix multiplication is associative
  7. Determinant of upper triangular matrix

Let $A$ and $B$ be $n$ by $n$ matrices. Then $|AB| = |A||B|$.

Proof

Case 1: $|A| = 0$

Assume $|AB| \neq 0$. \begin{align} & |AB| \neq 0 \\ &\Rightarrow \operatorname{rank}(AB) = n \\ &\Rightarrow (AB)^{-1} \textrm{ exists} \\ &\Rightarrow I = AB(AB)^{-1} \textrm{ exists} \\ &\Rightarrow B(AB)^{-1} \textrm{ is a right inverse of } A \\ &\Rightarrow A \textrm{ is invertible} \\ &\Rightarrow \operatorname{rank}(A) = n \\ &\Rightarrow |A| \neq 0 \\ &\Rightarrow \bot \end{align}

Therefore, $|AB| = 0 = |A||B|$.

Case 2: $|A| \neq 0$

\[ |A| \neq 0 \iff \operatorname{rank}(A) = n \iff \operatorname{RREF}(A) = I \]

Therefore, $A$ is row-equivalent to $I$. Let $[R_1, R_2, \ldots, R_p]$ be the sequence of matrices of the elementary row operations which convert $I$ to $A$. Let $[r_1, r_2, \ldots, r_p]$ be the determinant constants associated with those operations.

\begin{align} |AB| &= |R_p(R_{p-1}(\ldots(R_1(I))\ldots)) B| \\ &= |R_p(R_{p-1}(\ldots(R_1(B))\ldots))| \tag{Matrix multiplication is associative} \\ &= r_p(r_{p-1}(\ldots(r_1|B|)\ldots)) \\ &= (r_pr_{p-1}\ldots r_1)|B| \\ &= (r_pr_{p-1}\ldots r_1)|I||B| \tag{$I$ is upper-triangular, so $|I| = 1$} \\ &= (r_p(r_{p-1}(\ldots(r_1|I|)\ldots)))|B| \\ &= |(R_p(R_{p-1}(\ldots(R_1(I))\ldots)))||B| \\ &= |A||B| \end{align}

Dependency for:

  1. Every complex matrix has an eigenvalue

Info:

Transitive dependencies:

  1. /linear-algebra/vector-spaces/condition-for-subspace
  2. /linear-algebra/matrices/gauss-jordan-algo
  3. /sets-and-relations/equivalence-relation
  4. Group
  5. Ring
  6. Polynomial
  7. Integral Domain
  8. Comparing coefficients of a polynomial with disjoint variables
  9. 0x = 0 = x0
  10. Field
  11. Vector Space
  12. Linear independence
  13. Span
  14. A field is an integral domain
  15. Semiring
  16. Matrix
  17. Stacking
  18. System of linear equations
  19. Product of stacked matrices
  20. Matrix multiplication is associative
  21. Reduced Row Echelon Form (RREF)
  22. Submatrix
  23. Determinant
  24. Determinant of upper triangular matrix
  25. Swapping last 2 rows of a matrix negates its determinant
  26. Matrices over a field form a vector space
  27. Row space
  28. Elementary row operation
  29. Determinant after elementary row operation
  30. Every elementary row operation has a unique inverse
  31. Row equivalence of matrices
  32. Row equivalent matrices have the same row space
  33. RREF is unique
  34. Identity matrix
  35. Inverse of a matrix
  36. Inverse of product
  37. Elementary row operation is matrix pre-multiplication
  38. Row equivalence matrix
  39. Equations with row equivalent matrices have the same solution set
  40. Rank of a matrix
  41. Basis of a vector space
  42. Linearly independent set is not bigger than a span
  43. Homogeneous linear equations with more variables than equations
  44. Rank of a homogenous system of linear equations
  45. Full-rank square matrix in RREF is the identity matrix
  46. A matrix is full-rank iff its determinant is non-0
  47. Full-rank square matrix is invertible
  48. AB = I implies BA = I