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Consider the matrices

$A=\begin{pmatrix} 0 & 0 & 0 & 0 & 15 \\ 0 & 0 & 0 & 13 &14 \\ 0 & 0 & 10 & 11 & 12 \\ 0 & 6 & 7 & 8 & 9 \\ 1 & 2 & 3 & 4 & 5 \end{pmatrix} \text{ and}\; B=\begin{pmatrix} 5 & 4 & 3 & 2 & 1 \\ 9 & 8 & 7 & 6 & 0 \\ 12 & 11 & 10 & 0 & 0 \\ 14 & 13 & 0 & 0 & 0 \\ 15 & 0 & 0 & 0 & 0 \end{pmatrix}$

Which of the following hold true?

  1. $|A|=|B|$
  2. $\text{trace}(A)=\text{trace}(B)$
  3. $|A|=-|B|$
  4. $\text{trace}(AB)=\text{trace}(BA)$
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Solution: (A), (B) and (D) are true. The given matrices are obtained from $\begin{pmatrix} 1&2&3&4&5 \\0&6&7&8&9\\0&0&10&11&12\\0&0&0&13&14\\0&0&0&0&15 \end{pmatrix}$ by $10$ row and$10$ column exchanges respectively.
Answer:

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