Prove that the following determinant equals $0$
$begingroup$
We have a $ntimes n$ matrix $A=(a_{i,j})$.
If $i=j$, then $a_{i,j}=1-n$. otherwise, $a_{i,j}=1$. Show that $|A|=0$.
I tried using gauss elimination but it just gets too complicated. I also tried to do $R_i rightarrow R_i-R_1/(1-n)$ for all $i>1$, and then expand over the first column, but it also didn't work for me. Can someone please help?
linear-algebra determinant
edited Jan 15 at 15:19
Asaf Karagila♦
309k33441775
asked Jan 15 at 11:00
OmerOmer
535110
$endgroup$
add a comment |
$begingroup$
We have a $ntimes n$ matrix $A=(a_{i,j})$.
If $i=j$, then $a_{i,j}=1-n$. otherwise, $a_{i,j}=1$. Show that $|A|=0$.
I tried using gauss elimination but it just gets too complicated. I also tried to do $R_i rightarrow R_i-R_1/(1-n)$ for all $i>1$, and then expand over the first column, but it also didn't work for me. Can someone please help?
linear-algebra determinant
edited Jan 15 at 15:19
Asaf Karagila♦
309k33441775
asked Jan 15 at 11:00
OmerOmer
535110
$endgroup$
1
$begingroup$
The matrix of all ones is often denoted $mathbf J$. Your problem concerns $mathbf J - nmathbf I$, and it has been discussed in several previous Questions.
$endgroup$
– hardmath
Jan 15 at 15:40
add a comment |
$begingroup$
We have a $ntimes n$ matrix $A=(a_{i,j})$.
If $i=j$, then $a_{i,j}=1-n$. otherwise, $a_{i,j}=1$. Show that $|A|=0$.
I tried using gauss elimination but it just gets too complicated. I also tried to do $R_i rightarrow R_i-R_1/(1-n)$ for all $i>1$, and then expand over the first column, but it also didn't work for me. Can someone please help?
linear-algebra determinant
edited Jan 15 at 15:19
Asaf Karagila♦
309k33441775
asked Jan 15 at 11:00
OmerOmer
535110
$endgroup$
We have a $ntimes n$ matrix $A=(a_{i,j})$.
If $i=j$, then $a_{i,j}=1-n$. otherwise, $a_{i,j}=1$. Show that $|A|=0$.
I tried using gauss elimination but it just gets too complicated. I also tried to do $R_i rightarrow R_i-R_1/(1-n)$ for all $i>1$, and then expand over the first column, but it also didn't work for me. Can someone please help?
linear-algebra determinant
linear-algebra determinant
edited Jan 15 at 15:19
Asaf Karagila♦
309k33441775
asked Jan 15 at 11:00
OmerOmer
535110
edited Jan 15 at 15:19
Asaf Karagila♦
309k33441775
asked Jan 15 at 11:00
OmerOmer
535110
edited Jan 15 at 15:19
Asaf Karagila♦
309k33441775
edited Jan 15 at 15:19
Asaf Karagila♦
309k33441775
edited Jan 15 at 15:19
Asaf Karagila♦
309k33441775
309k33441775
asked Jan 15 at 11:00
OmerOmer
535110
asked Jan 15 at 11:00
OmerOmer
535110
asked Jan 15 at 11:00
OmerOmer
535110
535110
1
$begingroup$
The matrix of all ones is often denoted $mathbf J$. Your problem concerns $mathbf J - nmathbf I$, and it has been discussed in several previous Questions.
$endgroup$
– hardmath
Jan 15 at 15:40
add a comment |
1
$begingroup$
The matrix of all ones is often denoted $mathbf J$. Your problem concerns $mathbf J - nmathbf I$, and it has been discussed in several previous Questions.
$endgroup$
– hardmath
Jan 15 at 15:40
1
1
$begingroup$
The matrix of all ones is often denoted $mathbf J$. Your problem concerns $mathbf J - nmathbf I$, and it has been discussed in several previous Questions.
$endgroup$
– hardmath
Jan 15 at 15:40
$begingroup$
The matrix of all ones is often denoted $mathbf J$. Your problem concerns $mathbf J - nmathbf I$, and it has been discussed in several previous Questions.
$endgroup$
– hardmath
Jan 15 at 15:40
add a comment |
1 Answer
1
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oldest
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$begingroup$
Hint:$$
Aleft(begin{array}{c}1\1\vdots\1end{array}right)=0.
$$
answered Jan 15 at 11:15
SongSong
18.6k21651
$endgroup$
add a comment |
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$begingroup$
Hint:$$
Aleft(begin{array}{c}1\1\vdots\1end{array}right)=0.
$$
answered Jan 15 at 11:15
SongSong
18.6k21651
$endgroup$
add a comment |
$begingroup$
Hint:$$
Aleft(begin{array}{c}1\1\vdots\1end{array}right)=0.
$$
answered Jan 15 at 11:15
SongSong
18.6k21651
$endgroup$
add a comment |
$begingroup$
Hint:$$
Aleft(begin{array}{c}1\1\vdots\1end{array}right)=0.
$$
answered Jan 15 at 11:15
SongSong
18.6k21651
$endgroup$
Hint:$$
Aleft(begin{array}{c}1\1\vdots\1end{array}right)=0.
$$
answered Jan 15 at 11:15
SongSong
18.6k21651
answered Jan 15 at 11:15
SongSong
18.6k21651
answered Jan 15 at 11:15
SongSong
18.6k21651
answered Jan 15 at 11:15
SongSong
18.6k21651
18.6k21651
add a comment |
add a comment |
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1
$begingroup$
The matrix of all ones is often denoted $mathbf J$. Your problem concerns $mathbf J - nmathbf I$, and it has been discussed in several previous Questions.
$endgroup$
– hardmath
Jan 15 at 15:40