n by n matrix

Apr 2010
30
0
can someone please give me an example of an n x n matrix, where the last column is the sum of the previous n-1 columns?
i have to prove whether it is invertible or not

thanks
 
Nov 2009
485
184
Is there something preventing you from doing this on your own?
 
Apr 2010
30
0
i'm not sure whether the example i made up was correct

a11 a12 a13 | (a13+ a23+ a33)
a21 a22 a23 | (a12 + a22 +a32)
a31 a32 a33 | (a11 + a21 +a31)

is this correct?
 
Nov 2009
485
184
That's not an \(\displaystyle n\times n\) matrix, though.
 
Apr 2010
30
0
woops. sorry i didn't see that :(

a11 a12 (a12:a32)
a21 a22 (a11:a31)
a31 a32 (...)

i get stuck when it comes to a33.
 
Sep 2009
41
0
i'm not sure whether the example i made up was correct

a11 a12 a13 | (a13+ a23+ a33)
a21 a22 a23 | (a12 + a22 +a32)
a31 a32 a33 | (a11 + a21 +a31)

is this correct?
i think you should add an extra row
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
can someone please give me an example of an n x n matrix, where the last column is the sum of the previous n-1 columns?
i have to prove whether it is invertible or not

thanks
To disprove something, all you need is an example. Since there aren't many stipulations on this matrix, I can think of many singular nxn matrices that meet your criteria.
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
Assume this is nxn

\(\displaystyle \begin{bmatrix}
a & a & b & \dots & \sum_{x=1}^{n}a_{1x}\\
a & a & c & & \\
a & a & d & & \vdots\\
a & a & e & \ddots & \\
a & a & f & & \sum_{x=1}^{n}a_{nx}
\end{bmatrix}\)

This matrix isn't invertible since column 1 and 2 are lin. dep.
 
Last edited:
Apr 2010
30
0
Assume this is nxn

\(\displaystyle \begin{bmatrix}
a & a & b & \dots & \sum_{x=1}^{n}a_{1x}\\
a & a & c & & \\
a & a & d & & \vdots\\
a & a & e & \ddots & \\
a & a & f & & \sum_{y=1}^{n}\sum_{x=1}^{n}a_{yx}
\end{bmatrix}\)

This matrix isn't invertible since column 1 and 2 are lin. dep.

are b,c,d...f the sum of the previous columns?