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Repertoire Method Clarification Required ( Concrete Mathematics )
Is the set of natural numbers closed under subtraction?

Let $A,\ B,\ C,\ D \in \mathcal{M}_n(\mathbb{C})$. If $\operatorname{rank}\left( \begin{bmatrix} A &B \\ C &D \end{bmatrix}\right)=n$, prove that $\det(AD)=\det(BC)$.

- Equivalent definitions of isometry
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- Invertibility of $BA$
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- Prove that if $(v_1,\ldots,v_n)$ spans $V$, then so does the list $(v_1-v_2,v_2-v_3,\ldots,v_{n-1}-v_n,v_n).$
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- Linear algebra proof regarding matrices

Assume first that the first $n$ columns are independent. Then the last $n$ columns are linear combinations of the first $n$. Therefore we have the equality

$$\begin{bmatrix} A \\ C \end{bmatrix} \cdot V = \begin{bmatrix} B \\ D \end{bmatrix}$$

so we’re done.

Now, if the first $n$ columns are not linearly independent, both sides are $0$.

Obs: to understand what $V$ is: the first column of $V$ consists of the coefficients in the writing of the first column of $\begin{bmatrix} B \\ D \end{bmatrix}$ as a linear combination of the columns of $\begin{bmatrix} A \\ C \end{bmatrix}$.

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- Can someone explain these strange properties of $10, 11, 12$ and $13$?
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