 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/digital-systems-linear-algebra-introduction-to-formal-proof/4/d/ii/solution

Each elementary matrix is obtained by applying one invertible elementary row operation to the identity matrix. Its inverse is the elementary matrix for the reverse operation. A product of elementary matrices is therefore invertible and has full rank

n

Conversely, if

rank A = n

, Gaussian elimination has a pivot in every column and reduces

A

to the identity. Writing the successive row operations as left multiplication gives

E_{r} \dots E_{1} A = I

. Hence

A = E_{1}^{- 1} \dots E_{r}^{- 1}

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