Answer
The forces needed to produce a deflection of .04 at point 1,2,3 respectively are ;
$0 lb,-5lb,10lb$
Work Step by Step
Given the following D Matrix;
$D= \begin{bmatrix}.005&.002&.001\\.002.&.004&.002\\.001&.002&.005\end{bmatrix}$
We compute the stiffness matrix $D^{-1} $ as
$D^{-1}= \begin{bmatrix}250&-125&0\\-125&375&-125\\ 0 &-125&250\end{bmatrix}$
To find the forces needed to produce a deflection of .04 inch at point 3 only,
$\mathbf {y}= \begin{bmatrix}0\\0\\0.04\end{bmatrix}$
but $\mathbf {y}=D\mathbf {f}$
hence;
$\mathbf { f } = D ^ { - 1 } \mathbf { y }$
$\mathbf { f }= \begin{bmatrix}250&-125&0\\-125&375&-125\\ 0 &-125&250\end{bmatrix}\begin{bmatrix}0\\0\\0.04\end{bmatrix}$
$\mathbf { f }= \begin{bmatrix}250\times0+-125\times0+0\times0.04\\-125\times0+375\times0+-125\times0.04\\ 0 \times0+-125\times0+250\times0.04\end{bmatrix}=\begin{bmatrix}0\\-5\\10\end{bmatrix}$
The forces needed to produce a deflection of .04 at point 1,2,3 respectively are ;
$0 lb,-5lb,10lb$
