algebra

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edison (8935)

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The rank of a matrix is the maximum number of independent rows (or, the maximum number of independent columns).

1 2 0 3
1 7 3 0
0 0 4 8
2 4 0 6

Start with row 1: See if row 2, 3 or 4 are multiples. If so, cross that out.
Go to row 2: See if row 3 or 4 are multiples...
Finally compare row 3 and 4.

This is for 4 rows, but you can extend this to n rows.

Do the same for the columns.

The rank would be 3 in the example above because we crossed out a row, leaving 3. The last row is a multiple of the first.

A paradox is an argument that starts with apparently acceptable assumptions and leads by apparently valid deductions to an apparent contradiction. Since logic admits no contradictions, either the apparently acceptable assumptions are not acceptable, or the apparently valid deductions are not valid, or the apparent contradiction is not a contradiction. A paradox moves us to reexamine the argument until we find out what is wrong.

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