|AB|=|A||B| Proof 1: direct computation (The most elementary proof)
Let A=\begin{pmatrix}a&b\\c&d\end{pmatrix} and B=\begin{pmatrix}x&y\\z&t\end{pmatrix}. Then AB=\begin{pmatrix}ax+bz&ay+bt\\cx+dz&cy+dt\end{pmatrix}. We thus have \begin{align}|AB|&=(ax+bz)(cy+dt)-(ay+bt)(cx+dz)\\
&=acxy+adxt+bcyz+bdtz-acxy-bcxt-adyz-bdzt\\
&=(ad-bc)xt-(ad-bc)yz\\
&=(ad-bc)(xt-yz)\\
&=|A|\cdot |B| \quad \Box
\end{align} Proof 2: exterior algebra / wedge product
|AB|=e_1\wedge \cdots \wedge e^n=|A|(Be_1\wedge \cdots \wedge e^n)=|A|\cdot |B|(e_1\wedge \cdots \wedge e_n) \;\Box
Proof 3: elementary row operations
Suppose A is singular. Then |A|=0 \Rightarrow |A||B|=0. On the other hand, |AB|=0 since \text{rank}(AB)\leq\text{rank}(A). Thus |AB|=|A||B|=0.
Suppose A is invertible. Then A is a product of elementary matrices A=E_1E_2\cdots E_k. Thus AB=E_1E_2\cdots E_kB. Then |AB|=|E_1||E_2|\cdots|E_k||B|=|A||B|. \quad \Box
More to explore
More proofs
Proof by analysis -- fundamental algebra of algebra
No comments:
Post a Comment