Which of the following statements is correct?
$begingroup$
My attempt: I take the matrices which have trace $-1$ and determinant $1$.
As I have tried this question many times and I did not get any matrices which satisfy the conditions of option a), option b), or option c), so from my point of view none of the options are correct.
Any help is appreciated.
linear-algebra matrices
edited Sep 13 '17 at 21:01
aidangallagher4
6661313
$endgroup$
|
show 2 more comments
$begingroup$
My attempt: I take the matrices which have trace $-1$ and determinant $1$.
As I have tried this question many times and I did not get any matrices which satisfy the conditions of option a), option b), or option c), so from my point of view none of the options are correct.
Any help is appreciated.
linear-algebra matrices
edited Sep 13 '17 at 21:01
aidangallagher4
6661313
$endgroup$
$begingroup$
You could vastly improve this question by giving it a meaningful title and using correct spelling and capitalization in your text.
$endgroup$
– davidlowryduda♦
Sep 13 '17 at 20:17
$begingroup$
Hint : Use the Caley-Hamilton-theorem
$endgroup$
– Peter
Sep 13 '17 at 20:19
$begingroup$
pliz help me for title ,,i don't know how to use maths jax @mixedmaths
$endgroup$
– user469754
Sep 13 '17 at 20:19
1
$begingroup$
If A is a permutation matrix of order $3$, then it satisfies the conditions of (c), but not the conclusion.
$endgroup$
– G Tony Jacobs
Sep 13 '17 at 20:25
2
$begingroup$
Counterexample for $a)$ : $$pmatrix {0&0&1\0&1&0\1&0&0}$$
$endgroup$
– Peter
Sep 13 '17 at 20:29
|
show 2 more comments
$begingroup$
My attempt: I take the matrices which have trace $-1$ and determinant $1$.
As I have tried this question many times and I did not get any matrices which satisfy the conditions of option a), option b), or option c), so from my point of view none of the options are correct.
Any help is appreciated.
linear-algebra matrices
edited Sep 13 '17 at 21:01
aidangallagher4
6661313
$endgroup$
My attempt: I take the matrices which have trace $-1$ and determinant $1$.
As I have tried this question many times and I did not get any matrices which satisfy the conditions of option a), option b), or option c), so from my point of view none of the options are correct.
Any help is appreciated.
linear-algebra matrices
linear-algebra matrices
edited Sep 13 '17 at 21:01
aidangallagher4
6661313
edited Sep 13 '17 at 21:01
aidangallagher4
6661313
edited Sep 13 '17 at 21:01
aidangallagher4
6661313
edited Sep 13 '17 at 21:01
aidangallagher4
6661313
edited Sep 13 '17 at 21:01
aidangallagher4
6661313
6661313
asked Sep 13 '17 at 20:14
user469754
$begingroup$
You could vastly improve this question by giving it a meaningful title and using correct spelling and capitalization in your text.
$endgroup$
– davidlowryduda♦
Sep 13 '17 at 20:17
$begingroup$
Hint : Use the Caley-Hamilton-theorem
$endgroup$
– Peter
Sep 13 '17 at 20:19
$begingroup$
pliz help me for title ,,i don't know how to use maths jax @mixedmaths
$endgroup$
– user469754
Sep 13 '17 at 20:19
1
$begingroup$
If A is a permutation matrix of order $3$, then it satisfies the conditions of (c), but not the conclusion.
$endgroup$
– G Tony Jacobs
Sep 13 '17 at 20:25
2
$begingroup$
Counterexample for $a)$ : $$pmatrix {0&0&1\0&1&0\1&0&0}$$
$endgroup$
– Peter
Sep 13 '17 at 20:29
|
show 2 more comments
$begingroup$
You could vastly improve this question by giving it a meaningful title and using correct spelling and capitalization in your text.
$endgroup$
– davidlowryduda♦
Sep 13 '17 at 20:17
$begingroup$
Hint : Use the Caley-Hamilton-theorem
$endgroup$
– Peter
Sep 13 '17 at 20:19
$begingroup$
pliz help me for title ,,i don't know how to use maths jax @mixedmaths
$endgroup$
– user469754
Sep 13 '17 at 20:19
1
$begingroup$
If A is a permutation matrix of order $3$, then it satisfies the conditions of (c), but not the conclusion.
$endgroup$
– G Tony Jacobs
Sep 13 '17 at 20:25
2
$begingroup$
Counterexample for $a)$ : $$pmatrix {0&0&1\0&1&0\1&0&0}$$
$endgroup$
– Peter
Sep 13 '17 at 20:29
$begingroup$
You could vastly improve this question by giving it a meaningful title and using correct spelling and capitalization in your text.
$endgroup$
– davidlowryduda♦
Sep 13 '17 at 20:17
$begingroup$
You could vastly improve this question by giving it a meaningful title and using correct spelling and capitalization in your text.
$endgroup$
– davidlowryduda♦
Sep 13 '17 at 20:17
$begingroup$
Hint : Use the Caley-Hamilton-theorem
$endgroup$
– Peter
Sep 13 '17 at 20:19
$begingroup$
Hint : Use the Caley-Hamilton-theorem
$endgroup$
– Peter
Sep 13 '17 at 20:19
$begingroup$
pliz help me for title ,,i don't know how to use maths jax @mixedmaths
$endgroup$
– user469754
Sep 13 '17 at 20:19
$begingroup$
pliz help me for title ,,i don't know how to use maths jax @mixedmaths
$endgroup$
– user469754
Sep 13 '17 at 20:19
1
1
$begingroup$
If A is a permutation matrix of order $3$, then it satisfies the conditions of (c), but not the conclusion.
$endgroup$
– G Tony Jacobs
Sep 13 '17 at 20:25
$begingroup$
If A is a permutation matrix of order $3$, then it satisfies the conditions of (c), but not the conclusion.
$endgroup$
– G Tony Jacobs
Sep 13 '17 at 20:25
2
2
$begingroup$
Counterexample for $a)$ : $$pmatrix {0&0&1\0&1&0\1&0&0}$$
$endgroup$
– Peter
Sep 13 '17 at 20:29
$begingroup$
Counterexample for $a)$ : $$pmatrix {0&0&1\0&1&0\1&0&0}$$
$endgroup$
– Peter
Sep 13 '17 at 20:29
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Here are some matrices to think about.
$A = pmatrix {1\&1\&&-1}$
$A = pmatrix {cos frac {2pi}{3}& sin frac {2pi}{3}\-sin frac {2pi}{3}&cos frac {2pi}{3}}$
$A = pmatrix {1&&\&cos frac {2pi}{3}& sin frac {2pi}{3}\&-sin frac {2pi}{3}&cos frac {2pi}{3}}$
In all of the scenarios
$A^n = I$ says something about the eigenvalues of $A$
Then there is a further clause that restricts what those eigenvalues may be.
And finally, do all matrices that meet the previous 2 constraints meet the 3rd constraint.
Update.
$A = pmatrix {-frac 12& frac {sqrt 3}{2}\-frac {sqrt 3}{2}&-frac 12}$
$A^2 = pmatrix {-frac 12& -frac {sqrt 3}{2}\frac {sqrt 3}{2}&-frac 12}$
$A^2+A+I = 0$
answered Sep 13 '17 at 20:35
Doug MDoug M
45.4k31954
$endgroup$
$begingroup$
let me check whethear this matrix satisfied or not @ Doug M
$endgroup$
– user469754
Sep 13 '17 at 20:40
1
$begingroup$
ya ,,i have checked all ur matrices that mean none of the of option is correct ,,,Am i right? @ doug M
$endgroup$
– user469754
Sep 13 '17 at 20:54
$begingroup$
No, the first matrix is a counterexample that proves a) is false. It meets the criteria, but not the conclusion. Same thing for the 3rd matrix and c). The second matrix meets the criteria and the conclusion of b), but that is not enough for a proof. But these examples should give you enough to work with to prove b)
$endgroup$
– Doug M
Sep 13 '17 at 20:57
$begingroup$
can u liitle bit elaborate why option b is coorect,,?@ Doug M,,,,,,as i have check ur given matrix for optiob B) that is A^2 + A +I is not equal to zero..
$endgroup$
– user469754
Sep 13 '17 at 21:03
$begingroup$
Sure it does... I have put in an update.
$endgroup$
– Doug M
Sep 13 '17 at 21:13
|
show 1 more comment
Your Answer
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1 Answer
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1 Answer
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$begingroup$
Here are some matrices to think about.
$A = pmatrix {1\&1\&&-1}$
$A = pmatrix {cos frac {2pi}{3}& sin frac {2pi}{3}\-sin frac {2pi}{3}&cos frac {2pi}{3}}$
$A = pmatrix {1&&\&cos frac {2pi}{3}& sin frac {2pi}{3}\&-sin frac {2pi}{3}&cos frac {2pi}{3}}$
In all of the scenarios
$A^n = I$ says something about the eigenvalues of $A$
Then there is a further clause that restricts what those eigenvalues may be.
And finally, do all matrices that meet the previous 2 constraints meet the 3rd constraint.
Update.
$A = pmatrix {-frac 12& frac {sqrt 3}{2}\-frac {sqrt 3}{2}&-frac 12}$
$A^2 = pmatrix {-frac 12& -frac {sqrt 3}{2}\frac {sqrt 3}{2}&-frac 12}$
$A^2+A+I = 0$
answered Sep 13 '17 at 20:35
Doug MDoug M
45.4k31954
$endgroup$
$begingroup$
let me check whethear this matrix satisfied or not @ Doug M
$endgroup$
– user469754
Sep 13 '17 at 20:40
1
$begingroup$
ya ,,i have checked all ur matrices that mean none of the of option is correct ,,,Am i right? @ doug M
$endgroup$
– user469754
Sep 13 '17 at 20:54
$begingroup$
No, the first matrix is a counterexample that proves a) is false. It meets the criteria, but not the conclusion. Same thing for the 3rd matrix and c). The second matrix meets the criteria and the conclusion of b), but that is not enough for a proof. But these examples should give you enough to work with to prove b)
$endgroup$
– Doug M
Sep 13 '17 at 20:57
$begingroup$
can u liitle bit elaborate why option b is coorect,,?@ Doug M,,,,,,as i have check ur given matrix for optiob B) that is A^2 + A +I is not equal to zero..
$endgroup$
– user469754
Sep 13 '17 at 21:03
$begingroup$
Sure it does... I have put in an update.
$endgroup$
– Doug M
Sep 13 '17 at 21:13
|
show 1 more comment
$begingroup$
Here are some matrices to think about.
$A = pmatrix {1\&1\&&-1}$
$A = pmatrix {cos frac {2pi}{3}& sin frac {2pi}{3}\-sin frac {2pi}{3}&cos frac {2pi}{3}}$
$A = pmatrix {1&&\&cos frac {2pi}{3}& sin frac {2pi}{3}\&-sin frac {2pi}{3}&cos frac {2pi}{3}}$
In all of the scenarios
$A^n = I$ says something about the eigenvalues of $A$
Then there is a further clause that restricts what those eigenvalues may be.
And finally, do all matrices that meet the previous 2 constraints meet the 3rd constraint.
Update.
$A = pmatrix {-frac 12& frac {sqrt 3}{2}\-frac {sqrt 3}{2}&-frac 12}$
$A^2 = pmatrix {-frac 12& -frac {sqrt 3}{2}\frac {sqrt 3}{2}&-frac 12}$
$A^2+A+I = 0$
answered Sep 13 '17 at 20:35
Doug MDoug M
45.4k31954
$endgroup$
$begingroup$
let me check whethear this matrix satisfied or not @ Doug M
$endgroup$
– user469754
Sep 13 '17 at 20:40
1
$begingroup$
ya ,,i have checked all ur matrices that mean none of the of option is correct ,,,Am i right? @ doug M
$endgroup$
– user469754
Sep 13 '17 at 20:54
$begingroup$
No, the first matrix is a counterexample that proves a) is false. It meets the criteria, but not the conclusion. Same thing for the 3rd matrix and c). The second matrix meets the criteria and the conclusion of b), but that is not enough for a proof. But these examples should give you enough to work with to prove b)
$endgroup$
– Doug M
Sep 13 '17 at 20:57
$begingroup$
can u liitle bit elaborate why option b is coorect,,?@ Doug M,,,,,,as i have check ur given matrix for optiob B) that is A^2 + A +I is not equal to zero..
$endgroup$
– user469754
Sep 13 '17 at 21:03
$begingroup$
Sure it does... I have put in an update.
$endgroup$
– Doug M
Sep 13 '17 at 21:13
|
show 1 more comment
$begingroup$
Here are some matrices to think about.
$A = pmatrix {1\&1\&&-1}$
$A = pmatrix {cos frac {2pi}{3}& sin frac {2pi}{3}\-sin frac {2pi}{3}&cos frac {2pi}{3}}$
$A = pmatrix {1&&\&cos frac {2pi}{3}& sin frac {2pi}{3}\&-sin frac {2pi}{3}&cos frac {2pi}{3}}$
In all of the scenarios
$A^n = I$ says something about the eigenvalues of $A$
Then there is a further clause that restricts what those eigenvalues may be.
And finally, do all matrices that meet the previous 2 constraints meet the 3rd constraint.
Update.
$A = pmatrix {-frac 12& frac {sqrt 3}{2}\-frac {sqrt 3}{2}&-frac 12}$
$A^2 = pmatrix {-frac 12& -frac {sqrt 3}{2}\frac {sqrt 3}{2}&-frac 12}$
$A^2+A+I = 0$
answered Sep 13 '17 at 20:35
Doug MDoug M
45.4k31954
$endgroup$
Here are some matrices to think about.
$A = pmatrix {1\&1\&&-1}$
$A = pmatrix {cos frac {2pi}{3}& sin frac {2pi}{3}\-sin frac {2pi}{3}&cos frac {2pi}{3}}$
$A = pmatrix {1&&\&cos frac {2pi}{3}& sin frac {2pi}{3}\&-sin frac {2pi}{3}&cos frac {2pi}{3}}$
In all of the scenarios
$A^n = I$ says something about the eigenvalues of $A$
Then there is a further clause that restricts what those eigenvalues may be.
And finally, do all matrices that meet the previous 2 constraints meet the 3rd constraint.
Update.
$A = pmatrix {-frac 12& frac {sqrt 3}{2}\-frac {sqrt 3}{2}&-frac 12}$
$A^2 = pmatrix {-frac 12& -frac {sqrt 3}{2}\frac {sqrt 3}{2}&-frac 12}$
$A^2+A+I = 0$
answered Sep 13 '17 at 20:35
Doug MDoug M
45.4k31954
edited Sep 13 '17 at 21:13
answered Sep 13 '17 at 20:35
Doug MDoug M
45.4k31954
answered Sep 13 '17 at 20:35
Doug MDoug M
45.4k31954
answered Sep 13 '17 at 20:35
Doug MDoug M
45.4k31954
45.4k31954
$begingroup$
let me check whethear this matrix satisfied or not @ Doug M
$endgroup$
– user469754
Sep 13 '17 at 20:40
1
$begingroup$
ya ,,i have checked all ur matrices that mean none of the of option is correct ,,,Am i right? @ doug M
$endgroup$
– user469754
Sep 13 '17 at 20:54
$begingroup$
No, the first matrix is a counterexample that proves a) is false. It meets the criteria, but not the conclusion. Same thing for the 3rd matrix and c). The second matrix meets the criteria and the conclusion of b), but that is not enough for a proof. But these examples should give you enough to work with to prove b)
$endgroup$
– Doug M
Sep 13 '17 at 20:57
$begingroup$
can u liitle bit elaborate why option b is coorect,,?@ Doug M,,,,,,as i have check ur given matrix for optiob B) that is A^2 + A +I is not equal to zero..
$endgroup$
– user469754
Sep 13 '17 at 21:03
$begingroup$
Sure it does... I have put in an update.
$endgroup$
– Doug M
Sep 13 '17 at 21:13
|
show 1 more comment
$begingroup$
let me check whethear this matrix satisfied or not @ Doug M
$endgroup$
– user469754
Sep 13 '17 at 20:40
1
$begingroup$
ya ,,i have checked all ur matrices that mean none of the of option is correct ,,,Am i right? @ doug M
$endgroup$
– user469754
Sep 13 '17 at 20:54
$begingroup$
No, the first matrix is a counterexample that proves a) is false. It meets the criteria, but not the conclusion. Same thing for the 3rd matrix and c). The second matrix meets the criteria and the conclusion of b), but that is not enough for a proof. But these examples should give you enough to work with to prove b)
$endgroup$
– Doug M
Sep 13 '17 at 20:57
$begingroup$
can u liitle bit elaborate why option b is coorect,,?@ Doug M,,,,,,as i have check ur given matrix for optiob B) that is A^2 + A +I is not equal to zero..
$endgroup$
– user469754
Sep 13 '17 at 21:03
$begingroup$
Sure it does... I have put in an update.
$endgroup$
– Doug M
Sep 13 '17 at 21:13
$begingroup$
let me check whethear this matrix satisfied or not @ Doug M
$endgroup$
– user469754
Sep 13 '17 at 20:40
$begingroup$
let me check whethear this matrix satisfied or not @ Doug M
$endgroup$
– user469754
Sep 13 '17 at 20:40
1
1
$begingroup$
ya ,,i have checked all ur matrices that mean none of the of option is correct ,,,Am i right? @ doug M
$endgroup$
– user469754
Sep 13 '17 at 20:54
$begingroup$
ya ,,i have checked all ur matrices that mean none of the of option is correct ,,,Am i right? @ doug M
$endgroup$
– user469754
Sep 13 '17 at 20:54
$begingroup$
No, the first matrix is a counterexample that proves a) is false. It meets the criteria, but not the conclusion. Same thing for the 3rd matrix and c). The second matrix meets the criteria and the conclusion of b), but that is not enough for a proof. But these examples should give you enough to work with to prove b)
$endgroup$
– Doug M
Sep 13 '17 at 20:57
$begingroup$
No, the first matrix is a counterexample that proves a) is false. It meets the criteria, but not the conclusion. Same thing for the 3rd matrix and c). The second matrix meets the criteria and the conclusion of b), but that is not enough for a proof. But these examples should give you enough to work with to prove b)
$endgroup$
– Doug M
Sep 13 '17 at 20:57
$begingroup$
can u liitle bit elaborate why option b is coorect,,?@ Doug M,,,,,,as i have check ur given matrix for optiob B) that is A^2 + A +I is not equal to zero..
$endgroup$
– user469754
Sep 13 '17 at 21:03
$begingroup$
can u liitle bit elaborate why option b is coorect,,?@ Doug M,,,,,,as i have check ur given matrix for optiob B) that is A^2 + A +I is not equal to zero..
$endgroup$
– user469754
Sep 13 '17 at 21:03
$begingroup$
Sure it does... I have put in an update.
$endgroup$
– Doug M
Sep 13 '17 at 21:13
$begingroup$
Sure it does... I have put in an update.
$endgroup$
– Doug M
Sep 13 '17 at 21:13
|
show 1 more comment
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$begingroup$
You could vastly improve this question by giving it a meaningful title and using correct spelling and capitalization in your text.
$endgroup$
– davidlowryduda♦
Sep 13 '17 at 20:17
$begingroup$
Hint : Use the Caley-Hamilton-theorem
$endgroup$
– Peter
Sep 13 '17 at 20:19
$begingroup$
pliz help me for title ,,i don't know how to use maths jax @mixedmaths
$endgroup$
– user469754
Sep 13 '17 at 20:19
1
$begingroup$
If A is a permutation matrix of order $3$, then it satisfies the conditions of (c), but not the conclusion.
$endgroup$
– G Tony Jacobs
Sep 13 '17 at 20:25
2
$begingroup$
Counterexample for $a)$ : $$pmatrix {0&0&1\0&1&0\1&0&0}$$
$endgroup$
– Peter
Sep 13 '17 at 20:29