Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 11 / a / ii

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- Solution ii

Solution

Choose Cramer's rule, the permitted alternative to finding an inverse. The determinant is

Δ = 1 (2 - 1) - 2 (- 1 - 2) + 3 (1 + 4) = 22 \neq = 0.

Replacing each column by the right-hand side gives

Δ_{x} = det 1401 2 - 2 1 31 - 1 = 14 + 2 + 6 = 22,

Δ_{y} = det 1121401 31 - 1 = - 1 + 42 + 3 = 44,

Δ_{z} = det 112 2 - 2 1 1401 = - 2 - 2 + 70 = 66.

Consequently

x = Δ_{x} /Δ = 1, y = Δ_{y} /Δ = 2, z = Δ_{z} /Δ = 3.

The nonzero determinant also proves uniqueness.

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