 Solution

ID: 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/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.

 New to topics? Read the docs here!