 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/digital-systems-linear-algebra-introduction-to-formal-proof/5/a/iii/solution

Let

e_{ij}

denote the matrix unit with one at position

(i, j)

and zero elsewhere. The five inverses are

E_{1}^{- 1} = I + e_{21}, E_{2}^{- 1} = I + 2 e_{31}, E_{3}^{- 1} = I - e_{41}, E_{4}^{- 1} = I + e_{32}, E_{5}^{- 1} = I - e_{42} .

Thus each is the identity matrix with only the indicated off-diagonal matrix entry altered. These inverses reverse the corresponding row additions; no row scaling or permutation is involved.

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