Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Digital systems, linear algebra, introduction to formal proof / 4 / c

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- Solution c

Solution

For

i \neq = j

, the elementary row operation

R_{i} \leftarrow R_{i} - k R_{j}

leaves the determinant unchanged:

det (\dots, R_{i} - k R_{j}, \dots, R_{j}, \dots) = det A .

By multilinearity in row

i

, the left side is

det A

minus

k

times a determinant with row

j

appearing twice. That second determinant is zero by alternation. No row scaling or row interchange is involved.

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