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
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.
If they have to be, then yes. By last column, do you mean every column needs to be the sum? I thought you mean column n is the sum of all the columns.
It doesn't change anything then because column 2 is the sum of column 1 then; thus, column 1 and 2 are lin. ind.
What do you mean the last column is the sum of previous?
Does that mean the nth column is the sum or does that mean column two is the sum of column 1, column 3 is the sum of column 1 and 2....? If the second is the case, you can make the matrix columns the Fibonacci numbers.
1, 1, 2, 3, 5, 8, ....
*note $\displaystyle v_1$ means a $\displaystyle nx1$ column vector of the nxn matrix
If you make $\displaystyle v_1$ a column vector of all 1s, then $\displaystyle v_2$=$\displaystyle v_1$, $\displaystyle v_3$=$\displaystyle 2v_1$, $\displaystyle v_4$=$\displaystyle 5v_1$,....