Should the determinant of a $0times 0$ matrix be defined as $0$ or $1$? [duplicate]
This question already has an answer here:
What is the determinant of ? [closed]
2 answers
Is the 0x0 matrix (zero-times-zero matrix) a well-defined concept?
1 answer
Let $A$ be the $0times 0$ matrix and let $det(A)$ be its determinant. I am wondering if $det(A)$ should be defined as $0$ or $1$.
- If we use the definition that determinant of an $ntimes n$ matrix $(a_{ij})$ be defined as $displaystylesum(-1)^{tau(j_1cdots j_n)}a_{1j_1}cdots a_{n{j_n}}$, where $tau(j_1cdots j_n)$ is the inversion number of the permutation $j_1cdots j_n$, then since there is no term present, $det(A)$ should be defined as $0$.
- If we expand the $1times 1$ matrix $(1)$ along the first row, we obtain
$$1=det(1)=1cdotdet(A),$$ which implies that $det(A)$ should be defined as $1$.
Which definition of the determinant of the $0times 0$ matrix $A$, if any, makes more sense here?
definition determinant
asked 19 hours ago
Zuriel
1,5831028
marked as duplicate by Peter, Dietrich Burde, Ross Millikan, egreg, ccorn 16 hours ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
|
show 4 more comments
This question already has an answer here:
What is the determinant of ? [closed]
2 answers
Is the 0x0 matrix (zero-times-zero matrix) a well-defined concept?
1 answer
Let $A$ be the $0times 0$ matrix and let $det(A)$ be its determinant. I am wondering if $det(A)$ should be defined as $0$ or $1$.
- If we use the definition that determinant of an $ntimes n$ matrix $(a_{ij})$ be defined as $displaystylesum(-1)^{tau(j_1cdots j_n)}a_{1j_1}cdots a_{n{j_n}}$, where $tau(j_1cdots j_n)$ is the inversion number of the permutation $j_1cdots j_n$, then since there is no term present, $det(A)$ should be defined as $0$.
- If we expand the $1times 1$ matrix $(1)$ along the first row, we obtain
$$1=det(1)=1cdotdet(A),$$ which implies that $det(A)$ should be defined as $1$.
Which definition of the determinant of the $0times 0$ matrix $A$, if any, makes more sense here?
definition determinant
asked 19 hours ago
Zuriel
1,5831028
marked as duplicate by Peter, Dietrich Burde, Ross Millikan, egreg, ccorn 16 hours ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
3
Honest nonsnarky question: why does it matter?
– Randall
19 hours ago
The definition of a determinant would give an empty sum, which conventionally is $0$.
– Oscar Lanzi
19 hours ago
Some arguments here: math.stackexchange.com/questions/1762537/… and here: math.stackexchange.com/questions/1913141/…
– Martin R
19 hours ago
2
This is very similar to the question as to whether $0^0$ should be zero or one.
– amsmath
18 hours ago
2
@Peter Because when we solve this problem for a $0times0$ matrix, we can move to $-1times-1$ matrix :)
– Oldboy
17 hours ago
|
show 4 more comments
This question already has an answer here:
What is the determinant of ? [closed]
2 answers
Is the 0x0 matrix (zero-times-zero matrix) a well-defined concept?
1 answer
Let $A$ be the $0times 0$ matrix and let $det(A)$ be its determinant. I am wondering if $det(A)$ should be defined as $0$ or $1$.
- If we use the definition that determinant of an $ntimes n$ matrix $(a_{ij})$ be defined as $displaystylesum(-1)^{tau(j_1cdots j_n)}a_{1j_1}cdots a_{n{j_n}}$, where $tau(j_1cdots j_n)$ is the inversion number of the permutation $j_1cdots j_n$, then since there is no term present, $det(A)$ should be defined as $0$.
- If we expand the $1times 1$ matrix $(1)$ along the first row, we obtain
$$1=det(1)=1cdotdet(A),$$ which implies that $det(A)$ should be defined as $1$.
Which definition of the determinant of the $0times 0$ matrix $A$, if any, makes more sense here?
definition determinant
asked 19 hours ago
Zuriel
1,5831028
This question already has an answer here:
What is the determinant of ? [closed]
2 answers
Is the 0x0 matrix (zero-times-zero matrix) a well-defined concept?
1 answer
Let $A$ be the $0times 0$ matrix and let $det(A)$ be its determinant. I am wondering if $det(A)$ should be defined as $0$ or $1$.
- If we use the definition that determinant of an $ntimes n$ matrix $(a_{ij})$ be defined as $displaystylesum(-1)^{tau(j_1cdots j_n)}a_{1j_1}cdots a_{n{j_n}}$, where $tau(j_1cdots j_n)$ is the inversion number of the permutation $j_1cdots j_n$, then since there is no term present, $det(A)$ should be defined as $0$.
- If we expand the $1times 1$ matrix $(1)$ along the first row, we obtain
$$1=det(1)=1cdotdet(A),$$ which implies that $det(A)$ should be defined as $1$.
Which definition of the determinant of the $0times 0$ matrix $A$, if any, makes more sense here?
This question already has an answer here:
What is the determinant of ? [closed]
2 answers
Is the 0x0 matrix (zero-times-zero matrix) a well-defined concept?
1 answer
definition determinant
definition determinant
asked 19 hours ago
Zuriel
1,5831028
asked 19 hours ago
Zuriel
1,5831028
asked 19 hours ago
Zuriel
1,5831028
asked 19 hours ago
Zuriel
1,5831028
asked 19 hours ago
Zuriel
1,5831028
1,5831028
marked as duplicate by Peter, Dietrich Burde, Ross Millikan, egreg, ccorn 16 hours ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Peter, Dietrich Burde, Ross Millikan, egreg, ccorn 16 hours ago
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
3
Honest nonsnarky question: why does it matter?
– Randall
19 hours ago
The definition of a determinant would give an empty sum, which conventionally is $0$.
– Oscar Lanzi
19 hours ago
Some arguments here: math.stackexchange.com/questions/1762537/… and here: math.stackexchange.com/questions/1913141/…
– Martin R
19 hours ago
2
This is very similar to the question as to whether $0^0$ should be zero or one.
– amsmath
18 hours ago
2
@Peter Because when we solve this problem for a $0times0$ matrix, we can move to $-1times-1$ matrix :)
– Oldboy
17 hours ago
|
show 4 more comments
3
Honest nonsnarky question: why does it matter?
– Randall
19 hours ago
The definition of a determinant would give an empty sum, which conventionally is $0$.
– Oscar Lanzi
19 hours ago
Some arguments here: math.stackexchange.com/questions/1762537/… and here: math.stackexchange.com/questions/1913141/…
– Martin R
19 hours ago
2
This is very similar to the question as to whether $0^0$ should be zero or one.
– amsmath
18 hours ago
2
@Peter Because when we solve this problem for a $0times0$ matrix, we can move to $-1times-1$ matrix :)
– Oldboy
17 hours ago
3
3
Honest nonsnarky question: why does it matter?
– Randall
19 hours ago
Honest nonsnarky question: why does it matter?
– Randall
19 hours ago
The definition of a determinant would give an empty sum, which conventionally is $0$.
– Oscar Lanzi
19 hours ago
The definition of a determinant would give an empty sum, which conventionally is $0$.
– Oscar Lanzi
19 hours ago
Some arguments here: math.stackexchange.com/questions/1762537/… and here: math.stackexchange.com/questions/1913141/…
– Martin R
19 hours ago
Some arguments here: math.stackexchange.com/questions/1762537/… and here: math.stackexchange.com/questions/1913141/…
– Martin R
19 hours ago
2
2
This is very similar to the question as to whether $0^0$ should be zero or one.
– amsmath
18 hours ago
This is very similar to the question as to whether $0^0$ should be zero or one.
– amsmath
18 hours ago
2
2
@Peter Because when we solve this problem for a $0times0$ matrix, we can move to $-1times-1$ matrix :)
– Oldboy
17 hours ago
@Peter Because when we solve this problem for a $0times0$ matrix, we can move to $-1times-1$ matrix :)
– Oldboy
17 hours ago
|
show 4 more comments
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3
Honest nonsnarky question: why does it matter?
– Randall
19 hours ago
The definition of a determinant would give an empty sum, which conventionally is $0$.
– Oscar Lanzi
19 hours ago
Some arguments here: math.stackexchange.com/questions/1762537/… and here: math.stackexchange.com/questions/1913141/…
– Martin R
19 hours ago
2
This is very similar to the question as to whether $0^0$ should be zero or one.
– amsmath
18 hours ago
2
@Peter Because when we solve this problem for a $0times0$ matrix, we can move to $-1times-1$ matrix :)
– Oldboy
17 hours ago