Complementary minor of a kernel of given full-rank matrix $A$ is equal to minor of $A$.

Clash Royale CLAN TAG#URR8PPP

up vote
0
down vote

favorite

Let $d leq n$, $d,n inmathbbN$. Suppose $beginpmatrixA & B endpmatrix,beginpmatrixC \ D endpmatrix$ be a block matrix with $A in Mat_mathbbZ^dtimes d,B in Mat_mathbbZ^dtimes (n-d), C in Mat_mathbbZ^dtimes (n-d), D in Mat_mathbbZ^(n-d)times (n-)d$ such that 1) $beginpmatrixA & B endpmatrix$ has rank $d$ and 2) $$beginpmatrixA & B endpmatrixbeginpmatrixC \ D endpmatrix = 0 $$
which is equivalent to say
$$AC + BD=0.$$
Also suppose that each columns of $beginpmatrixC \ D endpmatrix$ consists of a $mathbbZ$-basis of $ker beginpmatrixA & B endpmatrix$.
Then can we say $det(A) = det(D)$? It is mentioned in some book as a basic fact of linear algebra, but I don't know how to prove it. Any hint will be appreciated.

linear-algebra change-of-basis

share|cite|improve this question

edited Aug 6 at 3:43

asked Aug 6 at 2:57

user124697

607512

add a comment | 

up vote
0
down vote

favorite

Let $d leq n$, $d,n inmathbbN$. Suppose $beginpmatrixA & B endpmatrix,beginpmatrixC \ D endpmatrix$ be a block matrix with $A in Mat_mathbbZ^dtimes d,B in Mat_mathbbZ^dtimes (n-d), C in Mat_mathbbZ^dtimes (n-d), D in Mat_mathbbZ^(n-d)times (n-)d$ such that 1) $beginpmatrixA & B endpmatrix$ has rank $d$ and 2) $$beginpmatrixA & B endpmatrixbeginpmatrixC \ D endpmatrix = 0 $$
which is equivalent to say
$$AC + BD=0.$$
Also suppose that each columns of $beginpmatrixC \ D endpmatrix$ consists of a $mathbbZ$-basis of $ker beginpmatrixA & B endpmatrix$.
Then can we say $det(A) = det(D)$? It is mentioned in some book as a basic fact of linear algebra, but I don't know how to prove it. Any hint will be appreciated.

linear-algebra change-of-basis

share|cite|improve this question

edited Aug 6 at 3:43

asked Aug 6 at 2:57

user124697

607512

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Let $d leq n$, $d,n inmathbbN$. Suppose $beginpmatrixA & B endpmatrix,beginpmatrixC \ D endpmatrix$ be a block matrix with $A in Mat_mathbbZ^dtimes d,B in Mat_mathbbZ^dtimes (n-d), C in Mat_mathbbZ^dtimes (n-d), D in Mat_mathbbZ^(n-d)times (n-)d$ such that 1) $beginpmatrixA & B endpmatrix$ has rank $d$ and 2) $$beginpmatrixA & B endpmatrixbeginpmatrixC \ D endpmatrix = 0 $$
which is equivalent to say
$$AC + BD=0.$$
Also suppose that each columns of $beginpmatrixC \ D endpmatrix$ consists of a $mathbbZ$-basis of $ker beginpmatrixA & B endpmatrix$.
Then can we say $det(A) = det(D)$? It is mentioned in some book as a basic fact of linear algebra, but I don't know how to prove it. Any hint will be appreciated.

linear-algebra change-of-basis

share|cite|improve this question

edited Aug 6 at 3:43

asked Aug 6 at 2:57

user124697

607512

Let $d leq n$, $d,n inmathbbN$. Suppose $beginpmatrixA & B endpmatrix,beginpmatrixC \ D endpmatrix$ be a block matrix with $A in Mat_mathbbZ^dtimes d,B in Mat_mathbbZ^dtimes (n-d), C in Mat_mathbbZ^dtimes (n-d), D in Mat_mathbbZ^(n-d)times (n-)d$ such that 1) $beginpmatrixA & B endpmatrix$ has rank $d$ and 2) $$beginpmatrixA & B endpmatrixbeginpmatrixC \ D endpmatrix = 0 $$
which is equivalent to say
$$AC + BD=0.$$
Also suppose that each columns of $beginpmatrixC \ D endpmatrix$ consists of a $mathbbZ$-basis of $ker beginpmatrixA & B endpmatrix$.
Then can we say $det(A) = det(D)$? It is mentioned in some book as a basic fact of linear algebra, but I don't know how to prove it. Any hint will be appreciated.

linear-algebra change-of-basis

share|cite|improve this question

edited Aug 6 at 3:43

asked Aug 6 at 2:57

user124697

607512

share|cite|improve this question

share|cite|improve this question

edited Aug 6 at 3:43

edited Aug 6 at 3:43

edited Aug 6 at 3:43

asked Aug 6 at 2:57

user124697

607512

asked Aug 6 at 2:57

user124697

607512

asked Aug 6 at 2:57

user124697

607512

607512

add a comment | 

add a comment | 

1 Answer
1

active

oldest

votes

up vote
1
down vote



accepted

Take $d=1$ and $n=2$; let $A=3$, $B=0$, $C=0$, $D=1$. These satisfy the hypotheses, yet $det Anot= det D$.

share|cite|improve this answer

edited Aug 6 at 8:14

answered Aug 6 at 6:25

ancientmathematician

4,1381312

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thank you for very simple counterexample :)
  – user124697
  Aug 6 at 18:54
  

  
  
  
  

add a comment | 

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2873550%2fcomplementary-minor-of-a-kernel-of-given-full-rank-matrix-a-is-equal-to-minor%23new-answer', 'question_page');

);

Post as a guest

Name Email

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote



accepted

Take $d=1$ and $n=2$; let $A=3$, $B=0$, $C=0$, $D=1$. These satisfy the hypotheses, yet $det Anot= det D$.

share|cite|improve this answer

edited Aug 6 at 8:14

answered Aug 6 at 6:25

ancientmathematician

4,1381312

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thank you for very simple counterexample :)
  – user124697
  Aug 6 at 18:54
  

  
  
  
  

add a comment | 

up vote
1
down vote



accepted

Take $d=1$ and $n=2$; let $A=3$, $B=0$, $C=0$, $D=1$. These satisfy the hypotheses, yet $det Anot= det D$.

share|cite|improve this answer

edited Aug 6 at 8:14

answered Aug 6 at 6:25

ancientmathematician

4,1381312

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thank you for very simple counterexample :)
  – user124697
  Aug 6 at 18:54
  

  
  
  
  

add a comment | 

up vote
1
down vote



accepted

up vote
1
down vote



accepted

Take $d=1$ and $n=2$; let $A=3$, $B=0$, $C=0$, $D=1$. These satisfy the hypotheses, yet $det Anot= det D$.

share|cite|improve this answer

edited Aug 6 at 8:14

answered Aug 6 at 6:25

ancientmathematician

4,1381312

Take $d=1$ and $n=2$; let $A=3$, $B=0$, $C=0$, $D=1$. These satisfy the hypotheses, yet $det Anot= det D$.

share|cite|improve this answer

edited Aug 6 at 8:14

answered Aug 6 at 6:25

ancientmathematician

4,1381312

share|cite|improve this answer

share|cite|improve this answer

edited Aug 6 at 8:14

edited Aug 6 at 8:14

edited Aug 6 at 8:14

answered Aug 6 at 6:25

ancientmathematician

4,1381312

answered Aug 6 at 6:25

ancientmathematician

4,1381312

answered Aug 6 at 6:25

ancientmathematician

4,1381312

4,1381312

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thank you for very simple counterexample :)
  – user124697
  Aug 6 at 18:54
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thank you for very simple counterexample :)
  – user124697
  Aug 6 at 18:54
  

  
  
  
  

Thank you for very simple counterexample :)
– user124697
Aug 6 at 18:54

Thank you for very simple counterexample :)
– user124697
Aug 6 at 18:54

add a comment | 

 

draft saved

draft discarded

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2873550%2fcomplementary-minor-of-a-kernel-of-given-full-rank-matrix-a-is-equal-to-minor%23new-answer', 'question_page');

);

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Name Email

Name

Email

Name Email

Name

Email