 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/12/a/ii/solution

To invert the matrix, row-reduce the augmented matrix

(M ∣ I)

. Subtracting row one from rows two and three gives left rows

(0, - 4, 1)

and

(0, 0, 2)

, so the pivots are nonzero and back-substitution is possible. The resulting matrix inverse is

M^{- 1} = 3/8 1/8 - 1/2 1/4 - 1/4 0 3/8 1/8 1/2 .

Multiplying by

M

verifies the identity matrix. This inverse is not

M^{T}

; indeed

M^{T} M = 3 - 1 0 - 1 110 002 \neq = I .

Thus $M$ is not orthogonal.

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