Let A be 10x10 invertible matrix with real entries s.t sum of each row is 1. Then which of follwing is true:

  1. Sum of entries of each row of inverse of A is 1.
  2. Sum if entriez of each column of inverse of A is 1.
  3. Trace of inverse of A is non zero.

I tried by assuming A to be 10x10 indentity matrix. But if we take this, then all options 1,2,3 are true. But answer in only 1.what other way is there?

  • $\begingroup$ Hint: The restriction to $10 \times 10$ matrices is inessential. So, what happens for $2 \times 2$ matrices? $\endgroup$ – Travis Willse Jan 23 '15 at 6:33
  • $\begingroup$ Randomly taken 2x2 matrix removes option 2, but still 3 option holds $\endgroup$ – Foggy Jan 23 '15 at 6:38
  • 1
    $\begingroup$ Let $M$ be the matrix with all entries $1$. Note that $A$ satisfies $AM=M$. Now $M=A^{-1}AM=A^{-1}M$. What does it tell you? $\endgroup$ – SMM Jan 23 '15 at 6:43
  • $\begingroup$ $A$ is invertible. How can it be nilpotent? $\endgroup$ – Ofir Schnabel Jan 23 '15 at 8:52

1 is the only one that is always true! It is easy to find a counterexample to 2, and for 3, you can use the block diagonal self inverse matrix with 5 diagonal blocks the type: \begin{align*}\left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right)\end{align*} And finally to answer why 1 is true, try looking at the vector $j = (1,1,\dots,1)^T$.

$Mj = j$, and thus $j=M^{-1}Mj = M^{-1}j$.


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