Use Sage to do your computations for you.
See Appendix B.
Recall that only nilpotent matrices of maximum degree of nilpotency are similar to elementary nilpotent form.
Determine which of the following nilpotent matrices can be put in elementary nilpotent form.
For those that can,
use Procedure 32.4.2 to construct an invertible matrix \(P\) such that \(P^{-1} A P\) is in elementary nilpotent form.
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\( A = \left[\begin{array}{rrr} 18 & -21 & -9 \\ 18 & -21 & -9 \\ -6 & 7 & 3 \end{array}\right] \)
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\( A = \left[\begin{array}{rrr} 8 & 10 & 4 \\ 25 & 36 & 14 \\ -79 & -113 & -44 \end{array}\right] \)
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\( A = \left[\begin{array}{rccc} -6 & 3 & 1 & 0 \\ -12 & 6 & 2 & 0 \\ -2 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \end{array}\right] \)
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\( A = \left[\begin{array}{rrrr} 4 & 8 & 18 & 2 \\ -11 & -22 & -54 & -19 \\ 4 & 8 & 20 & 8 \\ -1 & -2 & -5 & -2 \end{array}\right] \)
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\( A = \left[\begin{array}{rrrr} -3 & 15 & -9 & 111 \\ 0 & 26 & 0 & 169 \\ 1 & -11 & 3 & -76 \\ 0 & -4 & 0 & -26 \end{array}\right] \)
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\( A = \left[\begin{array}{ccrcrr} 0 & 2 & -4 & 8 & -8 & 43 \\ 0 & 1 & -1 & 4 & -6 & 21 \\ 0 & 1 & -1 & 2 & -4 & 11 \\ 0 & 0 & 1 & 1 & -3 & 5 \\ 0 & 0 & 0 & 1 & -1 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \)
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\( A = \left[\begin{array}{rrrrrr} 0 & -3 & 1 & 1 & -2 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 13 & 2 & -2 & 2 & -4 \\ 0 & 2 & 1 & 0 & 0 & -2 \\ 0 & -4 & 1 & 1 & 0 & -2 \\ 0 & 6 & 1 & -1 & 1 & -2 \end{array}\right] \)
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