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/matrices/gauss-jordan-algo
  2. /linear-algebra/vector-spaces/condition-for-subspace
  3. /sets-and-relations/equivalence-relation
  4. Group
  5. Ring
  6. Polynomial
  7. Field
  8. Vector Space
  9. Linear independence
  10. Span
  11. 0x = 0 = x0
  12. Integral Domain
  13. Comparing coefficients of a polynomial with disjoint variables
  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. Swapping last 2 rows of a matrix negates its determinant
  25. Determinant of upper triangular matrix
  26. Elementary row operation
  27. Determinant after elementary row operation
  28. Every elementary row operation has a unique inverse
  29. Row equivalence of matrices
  30. Matrices over a field form a vector space
  31. Row space
  32. Row equivalent matrices have the same row space
  33. RREF is unique
  34. Identity matrix
  35. Full-rank square matrix in RREF is the identity matrix
  36. Inverse of a matrix
  37. Inverse of product
  38. Elementary row operation is matrix pre-multiplication
  39. Row equivalence matrix
  40. Equations with row equivalent matrices have the same solution set
  41. Rank of a matrix
  42. Basis of a vector space
  43. Linearly independent set is not bigger than a span
  44. Homogeneous linear equations with more variables than equations
  45. Rank of a homogenous system of linear equations
  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