# n by n matrix

#### Kbotz

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

#### roninpro

Is there something preventing you from doing this on your own?

#### Kbotz

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?

#### roninpro

That's not an $$\displaystyle n\times n$$ matrix, though.

#### Kbotz

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.

#### Dgphru

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
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
i think you should add an extra row
Adding an extra row makes the matrix 4x3 which isn't nxn

#### dwsmith

MHF Hall of Honor
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:

#### Kbotz

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?