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True or false: If $A^3 = 0$ , $A$ is not invertible. Justify answer. [duplicate]

If the matrix $A^3 = 0$, is $A$ singular?Invertible matricesProduct of matrices of different order is not invertiblecenter of invertible matricesvalues of $t inmathbb R$ the matrix is not invertibleTrue or False Question About Linear AlgebraProve $C = AB$ is not invertible.Invertible and non-invertible linear transformationHow to show that B is not invertibleWhy is this true for matrices? Linearly dependent columns $implies$ not invertibleWhy is matrix not invertible (row form)

0

1

$begingroup$

This question already has an answer here:

• 
  If the matrix $A^3 = 0$, is $A$ singular? [on hold]
  
  3 answers
  
  

can you help me with my homework:

If $A^3 = 0$ , is $A$ not invertible? and please justify why.

linear-algebra matrices

share|cite|improve this question

edited yesterday

Jan Villapaz

asked yesterday

Jan VillapazJan Villapaz

142

New contributor

Jan Villapaz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

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marked as duplicate by David Hill, YiFan, Arnaud D., loup blanc, John Omielan yesterday

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.

• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  What are your thoughts?
  $endgroup$
  – lulu
  yesterday
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  What hapens if $A$ was.
  $endgroup$
  – hamam_Abdallah
  yesterday
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Do you know any invariants that tell us whether a matrix is invertible or not?
  $endgroup$
  – bounceback
  yesterday
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  As you are new to the site: people here tend not to respond well, or at all, to questions like these that look like routine homework exercises and which show no effort. What have you tried? People will gladly meet you half way if you give us some sense of your attempts and where you are getting stuck.
  $endgroup$
  – lulu
  yesterday
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Duplicate: math.stackexchange.com/q/3138158/496634
  $endgroup$
  – YiFan
  yesterday
  

  
  
  
  

 | 
show 3 more comments

0

1

$begingroup$

This question already has an answer here:

• 
  If the matrix $A^3 = 0$, is $A$ singular? [on hold]
  
  3 answers
  
  

can you help me with my homework:

If $A^3 = 0$ , is $A$ not invertible? and please justify why.

linear-algebra matrices

share|cite|improve this question

edited yesterday

Jan Villapaz

asked yesterday

Jan VillapazJan Villapaz

142

New contributor

Jan Villapaz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

marked as duplicate by David Hill, YiFan, Arnaud D., loup blanc, John Omielan yesterday

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.

• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  What are your thoughts?
  $endgroup$
  – lulu
  yesterday
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  What hapens if $A$ was.
  $endgroup$
  – hamam_Abdallah
  yesterday
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Do you know any invariants that tell us whether a matrix is invertible or not?
  $endgroup$
  – bounceback
  yesterday
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  As you are new to the site: people here tend not to respond well, or at all, to questions like these that look like routine homework exercises and which show no effort. What have you tried? People will gladly meet you half way if you give us some sense of your attempts and where you are getting stuck.
  $endgroup$
  – lulu
  yesterday
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Duplicate: math.stackexchange.com/q/3138158/496634
  $endgroup$
  – YiFan
  yesterday
  

  
  
  
  

 | 
show 3 more comments

$begingroup$

This question already has an answer here:

• 
  If the matrix $A^3 = 0$, is $A$ singular? [on hold]
  
  3 answers
  
  

can you help me with my homework:

If $A^3 = 0$ , is $A$ not invertible? and please justify why.

linear-algebra matrices

share|cite|improve this question

edited yesterday

Jan Villapaz

asked yesterday

Jan VillapazJan Villapaz

142

New contributor

Jan Villapaz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited yesterday

Jan Villapaz

asked yesterday

Jan VillapazJan Villapaz

142

New contributor

Jan Villapaz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited yesterday

Jan Villapaz

asked yesterday

Jan VillapazJan Villapaz

142

New contributor

Jan Villapaz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked yesterday

Jan VillapazJan Villapaz

142

New contributor

Jan Villapaz is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked yesterday

Jan VillapazJan Villapaz

142

3 Answers
3

active

oldest

votes

3

$begingroup$

HINT:

$$det(MN) = det(M)det(N)$$

share|cite|improve this answer

answered yesterday

Fly by NightFly by Night

26k32978

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  That was a massive hint. Please give the OP time to respond to the comments
  $endgroup$
  – bounceback
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @bounceback Thanks for the advice, but I think I'm experienced enough to know how this site works: "Use comments to ask for more information or suggest improvements. Avoid answering questions in the comments."
  $endgroup$
  – Fly by Night
  yesterday
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I'm not talking about the site rules. I'm talking about maths pedagogy.
  $endgroup$
  – bounceback
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  A site for helping people with mathematics? You aren't making sense.
  $endgroup$
  – bounceback
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @FlybyNight thank you for the hint!
  $endgroup$
  – Jan Villapaz
  yesterday
  

  
  
  
  

 | 
show 2 more comments

2

$begingroup$

If a matrix $A$ satisfying

$A^3 = 0 tag 1$

were invertible, there would exist a matrix $B$ such that

$AB = BA = I; tag 2$

then

$A^2 = IA^2 = (BA)A^2 = B(A^3) = B(0) = 0; tag 3$

going a step further in the same direction yields

$A = IA = (BA)A = B(A^2) = B(0) = 0; tag 4$

but clearly the $0$ matrix is not invertible; thus the assumptions (1) and (2) are, taken together, inherently contradictory. Therefore no matrix satisfying (1) may also satisfy (2).

Nota Bene: In fact it is easily seen that this result generalizes to the case

$A^k = 0, ; text{some} ; k in Bbb N; tag 5$

that is, no invertible matrix satisfies (5). End of Note.

share|cite|improve this answer

edited yesterday

answered yesterday

Robert LewisRobert Lewis

47.8k23067

$endgroup$

add a comment | 

1

$begingroup$

A couple of methods of attack have already been proposed, Jan Villapaz, have they helped you?

Hamam Abdallah said "What happens if A was" (I presume be means "what happens if A was nverstible"). Suppose A has an inverse, A^{-1}. What happens if you multiply both sides of A^3= 0 repeatedly by A^{-1}?

"Fly by Night" and several other pointed out that det(AB)= det(A)det(B). If det(A^3)= det(0)= 0, what can you say about det(A)? And what about the invertibility of A?

share|cite|improve this answer

answered yesterday

user247327user247327

11.4k1516

$endgroup$

add a comment | 

3

$begingroup$

HINT:

$$det(MN) = det(M)det(N)$$

share|cite|improve this answer

answered yesterday

Fly by NightFly by Night

26k32978

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  That was a massive hint. Please give the OP time to respond to the comments
  $endgroup$
  – bounceback
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @bounceback Thanks for the advice, but I think I'm experienced enough to know how this site works: "Use comments to ask for more information or suggest improvements. Avoid answering questions in the comments."
  $endgroup$
  – Fly by Night
  yesterday
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I'm not talking about the site rules. I'm talking about maths pedagogy.
  $endgroup$
  – bounceback
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  A site for helping people with mathematics? You aren't making sense.
  $endgroup$
  – bounceback
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @FlybyNight thank you for the hint!
  $endgroup$
  – Jan Villapaz
  yesterday
  

  
  
  
  

 | 
show 2 more comments

3

$begingroup$

HINT:

$$det(MN) = det(M)det(N)$$

share|cite|improve this answer

answered yesterday

Fly by NightFly by Night

26k32978

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  That was a massive hint. Please give the OP time to respond to the comments
  $endgroup$
  – bounceback
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @bounceback Thanks for the advice, but I think I'm experienced enough to know how this site works: "Use comments to ask for more information or suggest improvements. Avoid answering questions in the comments."
  $endgroup$
  – Fly by Night
  yesterday
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I'm not talking about the site rules. I'm talking about maths pedagogy.
  $endgroup$
  – bounceback
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  A site for helping people with mathematics? You aren't making sense.
  $endgroup$
  – bounceback
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @FlybyNight thank you for the hint!
  $endgroup$
  – Jan Villapaz
  yesterday
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

HINT:

$$det(MN) = det(M)det(N)$$

share|cite|improve this answer

answered yesterday

Fly by NightFly by Night

26k32978

$endgroup$

share|cite|improve this answer

answered yesterday

Fly by NightFly by Night

26k32978

answered yesterday

Fly by NightFly by Night

26k32978

answered yesterday

Fly by NightFly by Night

26k32978

2

$begingroup$

If a matrix $A$ satisfying

$A^3 = 0 tag 1$

were invertible, there would exist a matrix $B$ such that

$AB = BA = I; tag 2$

then

$A^2 = IA^2 = (BA)A^2 = B(A^3) = B(0) = 0; tag 3$

going a step further in the same direction yields

$A = IA = (BA)A = B(A^2) = B(0) = 0; tag 4$

but clearly the $0$ matrix is not invertible; thus the assumptions (1) and (2) are, taken together, inherently contradictory. Therefore no matrix satisfying (1) may also satisfy (2).

Nota Bene: In fact it is easily seen that this result generalizes to the case

$A^k = 0, ; text{some} ; k in Bbb N; tag 5$

that is, no invertible matrix satisfies (5). End of Note.

share|cite|improve this answer

edited yesterday

answered yesterday

Robert LewisRobert Lewis

47.8k23067

$endgroup$

add a comment | 

2

$begingroup$

If a matrix $A$ satisfying

$A^3 = 0 tag 1$

were invertible, there would exist a matrix $B$ such that

$AB = BA = I; tag 2$

then

$A^2 = IA^2 = (BA)A^2 = B(A^3) = B(0) = 0; tag 3$

going a step further in the same direction yields

$A = IA = (BA)A = B(A^2) = B(0) = 0; tag 4$

but clearly the $0$ matrix is not invertible; thus the assumptions (1) and (2) are, taken together, inherently contradictory. Therefore no matrix satisfying (1) may also satisfy (2).

Nota Bene: In fact it is easily seen that this result generalizes to the case

$A^k = 0, ; text{some} ; k in Bbb N; tag 5$

that is, no invertible matrix satisfies (5). End of Note.

share|cite|improve this answer

edited yesterday

answered yesterday

Robert LewisRobert Lewis

47.8k23067

$endgroup$

add a comment | 

$begingroup$

If a matrix $A$ satisfying

$A^3 = 0 tag 1$

were invertible, there would exist a matrix $B$ such that

$AB = BA = I; tag 2$

then

$A^2 = IA^2 = (BA)A^2 = B(A^3) = B(0) = 0; tag 3$

going a step further in the same direction yields

$A = IA = (BA)A = B(A^2) = B(0) = 0; tag 4$

but clearly the $0$ matrix is not invertible; thus the assumptions (1) and (2) are, taken together, inherently contradictory. Therefore no matrix satisfying (1) may also satisfy (2).

Nota Bene: In fact it is easily seen that this result generalizes to the case

$A^k = 0, ; text{some} ; k in Bbb N; tag 5$

that is, no invertible matrix satisfies (5). End of Note.

share|cite|improve this answer

edited yesterday

answered yesterday

Robert LewisRobert Lewis

47.8k23067

$endgroup$

share|cite|improve this answer

edited yesterday

answered yesterday

Robert LewisRobert Lewis

47.8k23067

answered yesterday

Robert LewisRobert Lewis

47.8k23067

answered yesterday

Robert LewisRobert Lewis

47.8k23067

1

$begingroup$

A couple of methods of attack have already been proposed, Jan Villapaz, have they helped you?

Hamam Abdallah said "What happens if A was" (I presume be means "what happens if A was nverstible"). Suppose A has an inverse, A^{-1}. What happens if you multiply both sides of A^3= 0 repeatedly by A^{-1}?

"Fly by Night" and several other pointed out that det(AB)= det(A)det(B). If det(A^3)= det(0)= 0, what can you say about det(A)? And what about the invertibility of A?

share|cite|improve this answer

answered yesterday

user247327user247327

11.4k1516

$endgroup$

add a comment | 

1

$begingroup$

A couple of methods of attack have already been proposed, Jan Villapaz, have they helped you?

Hamam Abdallah said "What happens if A was" (I presume be means "what happens if A was nverstible"). Suppose A has an inverse, A^{-1}. What happens if you multiply both sides of A^3= 0 repeatedly by A^{-1}?

"Fly by Night" and several other pointed out that det(AB)= det(A)det(B). If det(A^3)= det(0)= 0, what can you say about det(A)? And what about the invertibility of A?

share|cite|improve this answer

answered yesterday

user247327user247327

11.4k1516

$endgroup$

add a comment | 

$begingroup$

A couple of methods of attack have already been proposed, Jan Villapaz, have they helped you?

Hamam Abdallah said "What happens if A was" (I presume be means "what happens if A was nverstible"). Suppose A has an inverse, A^{-1}. What happens if you multiply both sides of A^3= 0 repeatedly by A^{-1}?

"Fly by Night" and several other pointed out that det(AB)= det(A)det(B). If det(A^3)= det(0)= 0, what can you say about det(A)? And what about the invertibility of A?

share|cite|improve this answer

answered yesterday

user247327user247327

11.4k1516

$endgroup$

share|cite|improve this answer

answered yesterday

user247327user247327

11.4k1516

answered yesterday

user247327user247327

11.4k1516

answered yesterday

user247327user247327

11.4k1516