Sum of matrix elements stay the same after inversion

Question

I found something interesting but I'm not sure how to proof it.

Suppose we have a square Matrix $M$ which is invertible and each column sums up to $1$ with only positive elements. Then the sum of all elements of the inversion $M^{-1}$ will be the same as the sum of all elements in $M$.

Can someone help me why this holds?

score 1 · Accepted Answer · answered Nov 05 '18 at 15:26

1

The fact that each column sums to 1 tells us that if $v$ is the vector with all 1s as its entries, then $M^Tv=v$. This tells us that we also have $(M^{T})^{-1}v=v$, and since $(M^T)^{-1}=(M^{-1})^T$, we know that all the columns of $M^{-1}$ also sum to 1. In any matrix where each column sums to 1, the total sum of the entries in $M$ is $n$. Thus the two will have the same sum.

answered Nov 05 '18 at 15:26

jgon

28,469

First, thanks for your great input! How can I conclude from $M^Tv=v$ that $(M^T)^{-1}v=v$ holds? – Selos Nov 05 '18 at 16:46
1

Multiply both sides of the first equation by $(M^T)^{-1}$ and use the fact that a matrix times its inverse is the identity. – jgon Nov 05 '18 at 16:49
Thanks! I accepted your answer. – Selos Nov 05 '18 at 17:03

Sum of matrix elements stay the same after inversion

1 Answers1