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ISI2017-MMA-15

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The diagonal elements of a square matrix $M$ are odd integers while the off-diagonals are even integers. Then

  1. $M$ must be singular
  2. $M$ must be nonsingular
  3. There is not enough information to decide the singularity of $M$
  4. $M$ must have a positive eigenvalue.
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$A =  \begin{bmatrix} 1 &0 \\ 0 & 1 \end{bmatrix}$

determinant of A $\neq 0$

$B = \begin{bmatrix} 1 &0 &0 \\ 0& 1 &0 \\ 0& 0 &1 \end{bmatrix}$

determinant of  B $\neq 0$

$C = \begin{bmatrix} 3 & 2 \\ 4 & 3 \end{bmatrix}$

determinant of  C $\neq 0$

If we take any configuration with such condition, matrix will be always non-singular.

https://math.stackexchange.com/questions/2327643/non-singularity-of-a-square-matrix/2327667

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Always nonsingular(B)
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