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## Homework Statement

\begin{bmatrix}

3x_{1}& 7x_{2}& 4x_{3} \\

3x_{1}& 4x_{2}& 5x_{3} \\

x_{1}& 10x_{2}& 8x_{3} \\

8x_{1}& 8x_{2}& 6x_{3} \\

\end{bmatrix}

=

\begin{bmatrix}

26 \\

16 \\

33 \\

46 \\

\end{bmatrix}

the measurements represented by equations 1 and 3 above can be trusted more than those represented by equations 2 and 4 and are given twice the weight.

Write down an explicit matrix form for the system of equations.

Solve it using Matlab.

However all I really need is to find the weighting factor, I can do the rest from there, struggling to see how I can weight the first and third rows by a factor of two, whilst simultaneously leaving the first and fourth alone.

## Homework Equations

I am going to use

Ax = b

e = W(Ax-b)

so

e^{T}e = (Ax-b)^T*W^T*W*(Ax-b)

so A^T*W^T*W*A*x = A^T*W^T*W*b

basically multiply that out and solve via guassian elimination for x

## The Attempt at a Solution

e = W(Ax-b)

\begin{bmatrix}

e \\

e \\

e \\

e \\

\end{bmatrix}

=

\begin{bmatrix}

?& ?& ?& ?& \\

?& ?& ?& ?& \\

?& ?& ?& ?& \\

?& ?& ?& ?& \\

\end{bmatrix}

*

\begin{bmatrix}

3x_{1}& 7x_{2}& 4x_{3}& -26& \\

3x_{1}& 4x_{2}& 5x_{3}& -16& \\

x_{1}& 10x_{2}& 8x_{3}& -33& \\

8x_{1}& 8x_{2}& 6x_{3}& -46& \\

\end{bmatrix}

I have tried various combinations of 4x4 matrices for the ?????? matrix (weighting matrix) that will result in the weighting factor needed, e.g a diagonal 4x4 matrix of 2's, works however the other rows get multiplied by this as well. Please inform me of how to find this.