# The stiffness matrix of a beam is given as $K = \left[ {\begin{array}{*{20}{c}} {12}&4\\ 4&5 \end{array}} \right]$ Calculate the flexibility matrix

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The stiffness matrix of a beam is given as $K = \left[ {\begin{array}{*{20}{c}} {12}&4\\ 4&5 \end{array}} \right]$  Calculate the flexibility matrix. Flexibility matrix will be ______.
1. $\frac{K}{{44}}\left[ {\begin{array}{*{20}{c}} {12}&{ - 4}\\ { - 4}&5 \end{array}} \right]$
2. $\frac{K}{{44}}\left[ {\begin{array}{*{20}{c}} {12}&4\\ 4&5 \end{array}} \right]$
3. $\frac{1}{{44}}\left[ {\begin{array}{*{20}{c}} {12}&{ - 4}\\ { - 4}&5 \end{array}} \right]$
4. $\frac{1}{{44}}\left[ {\begin{array}{*{20}{c}} 5&{ - 4}\\ { - 4}&{12} \end{array}} \right]$

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Correct Answer - Option 4 : $\frac{1}{{44}}\left[ {\begin{array}{*{20}{c}} 5&{ - 4}\\ { - 4}&{12} \end{array}} \right]$

Explanation:

If we denote stiffness matrix as M and flexibility matrix as Δ

It is stiffness matrix, and then flexibility matrix is: Δ = K-1

Calculation:

Δ = $\frac{1}{{\left( {12\times5 - 4\times4} \right)}}\left[ {\begin{array}{*{20}{c}} 5&{ - 4}\\ { - 4}&{12} \end{array}} \right]$

∴ Δ = $\frac{1}{{44}}\left[ {\begin{array}{*{20}{c}} 5&{ - 4}\\ { - 4}&{12} \end{array}} \right]$