Proposition 4.5.2

• Save the files Prop452_minors.m, Prop452_3.txt, and Prop452_13.m to your computer.
• To show that 3 and 13 changes are necessary for ranks 3 and 1, respectively, we simply need to show that each 4x4 and 2x2 minor is nonzero. This is enough since, with 2 and 12 changes, there is always a 4x4 and 2x2 minor, respectively, left unchanged. To verify the minors are all nonzero, start Matlab and execute the command
        >   run('Prop452_minors.m')
  
• Here is the output displayed to the screen. Perusing it, we see that Matlab verified each minor is nonzero.
• To verify that 3 changes are sufficient to reduce to a rank 3 matrix, start Maple and execute the command
        >   read "Prop452_3.txt"
  
• Here is the output displayed to the screen. Perusing it, we see that Maple verified the rank is 3.
• To verify that 13 changes are sufficient to reduce to a rank 1 matrix, start Matlab and execute the command
        >   run('Prop452_13.m')
  
• Here is the output displayed to the screen. Perusing it, we see that Matlab verified the rank is 1 (which can also be done trivially by hand).