Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Digital systems, linear algebra, introduction to formal proof / 5 / b

Table of contents
- Solution b

Solution

Let

e_{i}

be standard coordinate columns. The required elementary matrix is

E = I_{n} + α e_{j} e_{i}^{T}, E A : R_{j} \leftarrow R_{j} + α R_{i} .

It is the identity matrix with the extra entry

α

(j, i)

. Left matrix multiplication implements the row operation: all other rows are unchanged. Since

j > i

,

E

is a lower triangular matrix with every diagonal entry one. Part (a) therefore gives

det (E) = 1

and

det (E A) = det (E) det (A) = det (A) .

Hence adding a multiple of one row to another leaves the determinant unchanged, even if

A

is singular.

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