determinant rule proof [closed]

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Can someone link me to proof of this rule? $$det beginbmatrix 0 & A & B\ C& D&0\ E&0&F endbmatrix =
det beginbmatrix 0 & A \ C& D endbmatrix times det beginbmatrix F endbmatrix + det beginbmatrix 0 & B\ E&F endbmatrix times det beginbmatrix D endbmatrix$$

determinant

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asked Jul 22 at 16:17

user8880707

62

closed as off-topic by John Ma, amWhy, Omnomnomnom, Isaac Browne, Claude Leibovici Jul 24 at 8:14

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – John Ma, amWhy, Omnomnomnom, Isaac Browne, Claude Leibovici

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Here's a good link to read.
  – amWhy
  Jul 22 at 16:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Are $A,B,...$ numbers or matrix blocks ?
  – Jean Marie
  Jul 22 at 16:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  they are matrix blocks
  – user8880707
  Jul 22 at 16:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This result is false (as tested on thousands of random cases). Unless the OP quickly rectifies it, this question should be closed.
  – Jean Marie
  Jul 22 at 16:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  math.stackexchange.com/a/2818434/559990
  – user8880707
  Jul 22 at 16:55
  

  
  
  
  

 | 
show 3 more comments

up vote
-5
down vote

favorite

Can someone link me to proof of this rule? $$det beginbmatrix 0 & A & B\ C& D&0\ E&0&F endbmatrix =
det beginbmatrix 0 & A \ C& D endbmatrix times det beginbmatrix F endbmatrix + det beginbmatrix 0 & B\ E&F endbmatrix times det beginbmatrix D endbmatrix$$

determinant

share|cite|improve this question

asked Jul 22 at 16:17

user8880707

62

closed as off-topic by John Ma, amWhy, Omnomnomnom, Isaac Browne, Claude Leibovici Jul 24 at 8:14

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – John Ma, amWhy, Omnomnomnom, Isaac Browne, Claude Leibovici

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Here's a good link to read.
  – amWhy
  Jul 22 at 16:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Are $A,B,...$ numbers or matrix blocks ?
  – Jean Marie
  Jul 22 at 16:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  they are matrix blocks
  – user8880707
  Jul 22 at 16:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This result is false (as tested on thousands of random cases). Unless the OP quickly rectifies it, this question should be closed.
  – Jean Marie
  Jul 22 at 16:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  math.stackexchange.com/a/2818434/559990
  – user8880707
  Jul 22 at 16:55
  

  
  
  
  

 | 
show 3 more comments

up vote
-5
down vote

favorite

up vote
-5
down vote

favorite

Can someone link me to proof of this rule? $$det beginbmatrix 0 & A & B\ C& D&0\ E&0&F endbmatrix =
det beginbmatrix 0 & A \ C& D endbmatrix times det beginbmatrix F endbmatrix + det beginbmatrix 0 & B\ E&F endbmatrix times det beginbmatrix D endbmatrix$$

determinant

share|cite|improve this question

asked Jul 22 at 16:17

user8880707

62

Can someone link me to proof of this rule? $$det beginbmatrix 0 & A & B\ C& D&0\ E&0&F endbmatrix =
det beginbmatrix 0 & A \ C& D endbmatrix times det beginbmatrix F endbmatrix + det beginbmatrix 0 & B\ E&F endbmatrix times det beginbmatrix D endbmatrix$$

determinant

share|cite|improve this question

asked Jul 22 at 16:17

user8880707

62

share|cite|improve this question

share|cite|improve this question

asked Jul 22 at 16:17

user8880707

62

asked Jul 22 at 16:17

user8880707

62

asked Jul 22 at 16:17

user8880707

62

62

closed as off-topic by John Ma, amWhy, Omnomnomnom, Isaac Browne, Claude Leibovici Jul 24 at 8:14

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – John Ma, amWhy, Omnomnomnom, Isaac Browne, Claude Leibovici

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by John Ma, amWhy, Omnomnomnom, Isaac Browne, Claude Leibovici Jul 24 at 8:14

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – John Ma, amWhy, Omnomnomnom, Isaac Browne, Claude Leibovici

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Here's a good link to read.
  – amWhy
  Jul 22 at 16:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Are $A,B,...$ numbers or matrix blocks ?
  – Jean Marie
  Jul 22 at 16:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  they are matrix blocks
  – user8880707
  Jul 22 at 16:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This result is false (as tested on thousands of random cases). Unless the OP quickly rectifies it, this question should be closed.
  – Jean Marie
  Jul 22 at 16:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  math.stackexchange.com/a/2818434/559990
  – user8880707
  Jul 22 at 16:55
  

  
  
  
  

 | 
show 3 more comments

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Here's a good link to read.
  – amWhy
  Jul 22 at 16:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Are $A,B,...$ numbers or matrix blocks ?
  – Jean Marie
  Jul 22 at 16:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  they are matrix blocks
  – user8880707
  Jul 22 at 16:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This result is false (as tested on thousands of random cases). Unless the OP quickly rectifies it, this question should be closed.
  – Jean Marie
  Jul 22 at 16:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  math.stackexchange.com/a/2818434/559990
  – user8880707
  Jul 22 at 16:55
  

  
  
  
  

1

1

Here's a good link to read.
– amWhy
Jul 22 at 16:21

Here's a good link to read.
– amWhy
Jul 22 at 16:21

Are $A,B,...$ numbers or matrix blocks ?
– Jean Marie
Jul 22 at 16:24

Are $A,B,...$ numbers or matrix blocks ?
– Jean Marie
Jul 22 at 16:24

they are matrix blocks
– user8880707
Jul 22 at 16:29

they are matrix blocks
– user8880707
Jul 22 at 16:29

This result is false (as tested on thousands of random cases). Unless the OP quickly rectifies it, this question should be closed.
– Jean Marie
Jul 22 at 16:53

This result is false (as tested on thousands of random cases). Unless the OP quickly rectifies it, this question should be closed.
– Jean Marie
Jul 22 at 16:53

math.stackexchange.com/a/2818434/559990
– user8880707
Jul 22 at 16:55

math.stackexchange.com/a/2818434/559990
– user8880707
Jul 22 at 16:55

 | 
show 3 more comments

1 Answer
1

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Here is a counterexample among many others for $2 times 2$ blocks with $0-1$ entries :

$A=beginpmatrix
0 & 0 & 1 & 0 & 0 & 1\
0 & 0 & 0 & 1 & 0 & 0\
0 & 1 & 1 & 0 & 0 & 0\
1 & 1 & 0 & 0 & 0 & 0\
1 & 1 & 0 & 0 & 1 & 0\
1 & 0 & 0 & 0 & 1 & 1
endpmatrix$

one gets $det(A)=0$ (proof : col. 1 + col. 3 = col. 2 + col. 6) whereas the second expression gives $-1$.

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answered Jul 22 at 17:16

Jean Marie

27.8k41847

add a comment | 

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote

Here is a counterexample among many others for $2 times 2$ blocks with $0-1$ entries :

$A=beginpmatrix
0 & 0 & 1 & 0 & 0 & 1\
0 & 0 & 0 & 1 & 0 & 0\
0 & 1 & 1 & 0 & 0 & 0\
1 & 1 & 0 & 0 & 0 & 0\
1 & 1 & 0 & 0 & 1 & 0\
1 & 0 & 0 & 0 & 1 & 1
endpmatrix$

one gets $det(A)=0$ (proof : col. 1 + col. 3 = col. 2 + col. 6) whereas the second expression gives $-1$.

share|cite|improve this answer

answered Jul 22 at 17:16

Jean Marie

27.8k41847

add a comment | 

up vote
1
down vote

Here is a counterexample among many others for $2 times 2$ blocks with $0-1$ entries :

$A=beginpmatrix
0 & 0 & 1 & 0 & 0 & 1\
0 & 0 & 0 & 1 & 0 & 0\
0 & 1 & 1 & 0 & 0 & 0\
1 & 1 & 0 & 0 & 0 & 0\
1 & 1 & 0 & 0 & 1 & 0\
1 & 0 & 0 & 0 & 1 & 1
endpmatrix$

one gets $det(A)=0$ (proof : col. 1 + col. 3 = col. 2 + col. 6) whereas the second expression gives $-1$.

share|cite|improve this answer

answered Jul 22 at 17:16

Jean Marie

27.8k41847

add a comment | 

up vote
1
down vote

up vote
1
down vote

Here is a counterexample among many others for $2 times 2$ blocks with $0-1$ entries :

$A=beginpmatrix
0 & 0 & 1 & 0 & 0 & 1\
0 & 0 & 0 & 1 & 0 & 0\
0 & 1 & 1 & 0 & 0 & 0\
1 & 1 & 0 & 0 & 0 & 0\
1 & 1 & 0 & 0 & 1 & 0\
1 & 0 & 0 & 0 & 1 & 1
endpmatrix$

one gets $det(A)=0$ (proof : col. 1 + col. 3 = col. 2 + col. 6) whereas the second expression gives $-1$.

share|cite|improve this answer

answered Jul 22 at 17:16

Jean Marie

27.8k41847

Here is a counterexample among many others for $2 times 2$ blocks with $0-1$ entries :

$A=beginpmatrix
0 & 0 & 1 & 0 & 0 & 1\
0 & 0 & 0 & 1 & 0 & 0\
0 & 1 & 1 & 0 & 0 & 0\
1 & 1 & 0 & 0 & 0 & 0\
1 & 1 & 0 & 0 & 1 & 0\
1 & 0 & 0 & 0 & 1 & 1
endpmatrix$

one gets $det(A)=0$ (proof : col. 1 + col. 3 = col. 2 + col. 6) whereas the second expression gives $-1$.

share|cite|improve this answer

answered Jul 22 at 17:16

Jean Marie

27.8k41847

share|cite|improve this answer

share|cite|improve this answer

answered Jul 22 at 17:16

Jean Marie

27.8k41847

answered Jul 22 at 17:16

Jean Marie

27.8k41847

answered Jul 22 at 17:16

Jean Marie

27.8k41847

27.8k41847

add a comment | 

add a comment |