 Solution

ID: 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/a/solution

R_{1} \leftarrow R_{1} - R_{2}

, which preserves the determinant. Expanding along the new first row gives

det A = det 2 - c c 8 0 c 7 0 c c = (2 - c) det (c 7 c c) = (2 - c) c (c - 7) .

A square matrix is an invertible matrix exactly when its determinant is nonzero. Therefore

c = 0, c = 2, c = 7

are precisely the three values making

A

a singular matrix. They also have direct explanations: at

c = 0

the second row is zero, at

c = 2

the first two rows coincide, and at

c = 7

the last two columns coincide.

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