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Let $A$ be an invertible $10\times 10$ matrix with real entries such that the sum of each row is $1$. Then choose the correct option.

  • The sum of the entries of each row of the inverse of $A$ is $1$.
  • The sum of the entries of each column of the inverse of $ A$ is $1$.
  • The trace of the inverse of $A$ is non-zero.
  • None of the above.

If the matrix is given we can find its inverse but how can we find its inverse if the matrix itself not given?

Any idea on how to find the answer?

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HINT:

What would happen if you multiplied this matrix by $\pmatrix{1 \\ 1 \\ \vdots \\ 1}$? What does this tell you?

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    $\begingroup$ We will get column matrix 10x1 with 1 in all the entries. $\endgroup$ – Sam Christopher May 30 '15 at 4:48
  • $\begingroup$ So what does that tell you? $\endgroup$ – user137731 May 30 '15 at 4:49
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    $\begingroup$ it is eigen vector and 1 is eigen value. $\endgroup$ – Sam Christopher May 30 '15 at 4:50
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    $\begingroup$ Good.. now you just need to use your knowledge of how the eigenvalues and eigenvectors of a matrix $A$ relate to the eigenvalues/vectors of the matrix $A^{-1}$. Then make some conclusions. I trust you can handle it from here. $\endgroup$ – user137731 May 30 '15 at 4:51

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