 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-1/17a/solution

Codex  0 Created 2026-09-23 Updated 2026-09-25

Ω^{[p, q]}

is the identity except in rows and columns

p, q

, where it has the block

(c - s s c), c = cos θ, s = sin θ .

For

a = A_{p j}

and

b = A_{q j}

, choose

c = \frac{a}{a ^{2} + b ^{2}}, s = \frac{b}{a ^{2} + b ^{2}}

when

(a, b) \neq = (0, 0)

. Then

(Ω^{[p, q]} A)_{q j} = - s a + c b = 0.

If both entries vanish, any angle works.

For the given matrix, first use

Ω^{[1, 3]} = 1/ 2 0 - 1/ 2 010 1/ 2 0 1/ 2 .

It gives

Ω^{[1, 3]} A = 200131103 .

Then use

Ω^{[2, 3]} = 100 0 3 /2 - 1/2 0 1/2 3 /2 .

Thus

R = Ω^{[2, 3]} Ω^{[1, 3]} A

R = 200120 1 3 /2 3/2 .

All leading row entries are positive. The resulting QR decomposition by Givens rotations is

A = QR,

where

Q = (Ω^{[2, 3]} Ω^{[1, 3]})^{T} = 2 /2 0 2 /2 - 2 /4 3 /2 2 /4 - 6 /4 - 1/2 6 /4

and

R

is the matrix above. Since it is a product of transposed rotations,

Q

is orthogonal.

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