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Results tagged with linear-algebra
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user 280363
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
4
votes
3
answers
359
views
Determinant in terms of certain $2\times 2$ minors
Let $A$ be an $n\times n$ matrix with entries $a_{i,j}$. Define an $(n-1)\times(n-1)$ matrix $B$ with entries $b_{i,j}=a_{1,1}a_{i+1,j+1}-a_{1,j+1}a_{i+1,1}$. Then $\det(B)=a_{1,1}^{n-2}\det(A)$.
I ca …
- 191
1
vote
Determinant in terms of certain $2\times 2$ minors
Thank you to @carlo-beenakker, @erick-wong, @quizzical, and @sam-hopkins for helpful answers and comments.
Investigating further, I found that the statement in the question appears to be a well-known …
- 191
6
votes
0
answers
374
views
Divisibility properties of minors of matrices
Let $A$ be an $m\times n$ matrix with integer entries. Let $d_i(A)$ be the greatest common divisor of all $i\times i$ minors of $A$, and define $d_0(A)=1$. Whenever $i\leq j$, one has that $d_i(A)$ di …
- 191