12-240/Classnotes for Tuesday November 15
From Drorbn
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Problem. Find the rank the matrix
Solution. Using (invertible!) row/column operations we aim to bring to look as close as possible to an identity matrix:
Do | Get |
1. Bring a to the upper left corner by swapping the first two rows and multiplying the first row (after the swap) by . | |
2. Add times the first row to the third row, in order to cancel the in position 3-1. | |
3. Likewise add times the first row to the fourth row, in order to cancel the in position 4-1. | |
4. With similar column operations (you need three of those) cancel all the entries in the first row (except, of course, the first, which is used in the canceling). | |
5. Turn the 2-2 entry to a by multiplying the second row by . | |
6. Using two row operations "clean" the second column; that is, cancel all entries in it other than the "pivot" at position 2-2. | |
7. Using three column operations clean the second row except the pivot. | |
8. Clean up the row and the column of the in position 3-3 by first multiplying the third row by and then performing the appropriate row and column transformations. Notice that by pure luck, the at position 4-5 of the matrix gets killed in action. |
Thus the rank of our matrix is 3.