 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/d/solution

Perform the elementary row operations from the bottom upwards:

R_{4} \leftarrow R_{4} - t R_{3}, R_{3} \leftarrow R_{3} - t R_{2}, R_{2} \leftarrow R_{2} - t R_{1} .

This order matters because each subtraction uses the predecessor's original row. For example,

(t^{3}, t^{2}, t, 1) - t (t^{2}, t, 1, t) = (0, 0, 0, 1 - t^{2}) .

The other two differences are

(0, 0, 1 - t^{2}, t - t^{3})

and

(0, 1 - t^{2}, t - t^{3}, t^{2} - t^{4})

U = 1000 t 1 - t^{2} 00 t^{2} t (1 - t^{2}) 1 - t^{2} 0 t^{3} t^{2} (1 - t^{2}) t (1 - t^{2}) 1 - t^{2} .

These elementary row operations use no division, so the reduction remains valid for all real numbers

t

, including

t = \pm 1

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