I wonder if anyone can share wrong but useful and/or fun proofs in linear algebra. It can not only be fun, but also useful for someone who learns the subject. You are welcome to add explanations but please hide them in order not to spoil the fun.
Let me start from my own "proof" that every square matrix has zero determinant (I am sure that "proof" was discovered many many times).
"Theorem". Every square matrix $A$ over any field $K$ has zero determinant.
"Proof". If $A$ has two equal rows or columns we are done. Otherwise, we will construct them. First let add all the rows except the first row to the first row. Next let add all the rows except the last row to the last row. Clearly, now first and last row are equal since both or them are sum of every row in $A$, so $\det(A) = 0$. $\Box$
After we added rows to the first row of $A$ it changed, but the "proof" assumes that the first row is the same.
Best Answer
I personally like the following wrong "proof" of the Cayley-Hamilton theorem (which is actually a true statement, just the proof is wrong).
Theorem: (Cayley-Hamilton). Let $K$ be a field and $A \in M_n(K)$ be an $n \times n$-matrix with entries in $K$. Then $\mathrm{charpol}_A(A)=0$ (this is $A$ is a root of its own characteristic polynomial).
"Proof": By definition $\mathrm{charpol}_A(x)= \det(xI_n-A)$ so plugging in $x=A$ we obtain $\mathrm{charpol}_A(A) = \det(AI_n-A) = \det(A-A) = \det(0_{n \times n}) = 0$. $\square$
What is wrong with this proof?