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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 5C / Solution

Let the columns of

A

be

a_{1}, a_{2}, a_{3}

. The Gram-Schmidt process gives

q_{1} = \frac{a _{1}}{∣ a _{1} ∣} = \frac{1}{2} (1, 1, 1, - 1)^{T}, r_{11} = 4.

Next,

r_{12} = q_{1}^{T} a_{2} = 2, a_{2} - r_{12} q_{1} = (1, - 1, 1, 1)^{T},

so

q_{2} = \frac{1}{2} (1, - 1, 1, 1)^{T}, r_{22} = 2.

For the final column,

r_{13} = q_{1}^{T} a_{3} = - 2, r_{23} = q_{2}^{T} a_{3} = 4,

and

a_{3} - r_{13} q_{1} - r_{23} q_{2} = (- 1, - 1, 1, - 1)^{T} .

Thus

q_{3} = \frac{1}{2} (- 1, - 1, 1, - 1)^{T}

and

r_{33} = 2

. The reduced QR decomposition is

Q = \frac{1}{2} 111 - 1 1 - 1 11 - 1 - 1 1 - 1, R = 400220 - 2 42 .

The columns of

Q

are orthonormal, and direct multiplication gives

QR = A

.

Solved by gpt-5.6-sol high.

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