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Chapter 2: Q42E (page 93)

(M) Let \(D\) as in Exercise 41, determine the forces that produce a deflection of .24 cm at the second point on the beam, with zero deflections at the other three points. How is the answer related to the enteries in \({D^{ - {\bf{1}}}}\)? (Hint: First answer the question when the deflection is 1 cm at the second point.)

Short Answer

Expert verified

\(\left( { - 104,\,167,\, - 113,\,56} \right)\;\;{\rm{newtons}}\)

The forces are 0.24 times the second column of \({D^{ - 1}}\).

Step by step solution

01

Find the inverse of matrix D

So, the inverse of the flexibility matrix is

\({D^{ - 1}} = \left( {\begin{aligned}{*{20}{c}}{533.333}&{ - 433.333}&{233.333}&{ - 133.333}\\{ - 433.333}&{695.833}&{ - 470.833}&{233.333}\\{233.333}&{ - 470.833}&{695.833}&{ - 433.333}\\{ - 133.333}&{233.333}&{ - 433.333}&{533.333}\end{aligned}} \right)\).

02

Solve the equation \(f = {D^{ - 1}}y\)

The deflection matrix is \(\left( {\begin{aligned}{*{20}{c}}0\\{.24}\\0\\0\end{aligned}} \right)\).

Use the equation \(f = {D^{ - 1}}y\).

\(\begin{aligned}{c}f = \left( {\begin{aligned}{*{20}{c}}{533.333}&{ - 433.333}&{233.333}&{ - 133.333}\\{ - 433.333}&{695.833}&{ - 470.833}&{233.333}\\{233.333}&{ - 470.833}&{695.833}&{ - 433.333}\\{ - 133.333}&{233.333}&{ - 433.333}&{533.333}\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}0\\{.24}\\0\\0\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{ - 104}\\{167}\\{ - 113}\\{56}\end{aligned}} \right)\end{aligned}\)

So, the forces required to produce deflections of \(\left( {0,\,\,0.24,\,0,\,0} \right)\;{\rm{cm}}\) at points 1, 2, 3, and 4, respectively, are \(\left( { - 104,\,167,\, - 113,\,56} \right)\;\;{\rm{newtons}}\).

03

Interpret the applied force

Forces required to produce zero deflection at all points except point 2 are 0.24 times of the second column of the inverse flexibility matrix.

As \(y \mapsto {D^{ - 1}}y\) represents the linear transformation, the forces that produce the deflection of 0.24 cm at the second point are 0.24 times the forces required to produce a deflection of 1 cm at the second point.

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Most popular questions from this chapter

In Exercises 1–9, assume that the matrices are partitioned conformably for block multiplication. In Exercises 5–8, find formulas for X, Y, and Zin terms of A, B, and C, and justify your calculations. In some cases, you may need to make assumptions about the size of a matrix in order to produce a formula. [Hint:Compute the product on the left, and set it equal to the right side.]

8. \[\left[ {\begin{array}{*{20}{c}}A&B\\{\bf{0}}&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}X&Y&Z\\{\bf{0}}&{\bf{0}}&I\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}I&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&I\end{array}} \right]\]

Describe in words what happens when you compute \({A^{\bf{5}}}\), \({A^{{\bf{10}}}}\), \({A^{{\bf{20}}}}\), and \({A^{{\bf{30}}}}\) for \(A = \left( {\begin{aligned}{*{20}{c}}{1/6}&{1/2}&{1/3}\\{1/2}&{1/4}&{1/4}\\{1/3}&{1/4}&{5/12}\end{aligned}} \right)\).

Exercises 15 and 16 concern arbitrary matrices A, B, and Cfor which the indicated sums and products are defined. Mark each statement True or False. Justify each answer.

15. a. If A and B are \({\bf{2}} \times {\bf{2}}\) with columns \({{\bf{a}}_1},{{\bf{a}}_2}\) and \({{\bf{b}}_1},{{\bf{b}}_2}\) respectively, then \(AB = \left( {\begin{aligned}{*{20}{c}}{{{\bf{a}}_1}{{\bf{b}}_1}}&{{{\bf{a}}_2}{{\bf{b}}_2}}\end{aligned}} \right)\).

b. Each column of ABis a linear combination of the columns of Busing weights from the corresponding column of A.

c. \(AB + AC = A\left( {B + C} \right)\)

d. \({A^T} + {B^T} = {\left( {A + B} \right)^T}\)

e. The transpose of a product of matrices equals the product of their transposes in the same order.

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation, and let Sand U be functions from \({\mathbb{R}^n}\) into \({\mathbb{R}^n}\) such that \(S\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) and \(\)\(U\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\). Show that \(U\left( v \right) = S\left( v \right)\) for all v in \({\mathbb{R}^n}\). This will show that Thas a unique inverse, as asserted in theorem 9. [Hint: Given any v in \({\mathbb{R}^n}\), we can write \({\mathop{\rm v}\nolimits} = T\left( {\mathop{\rm x}\nolimits} \right)\) for some x. Why? Compute \(S\left( {\mathop{\rm v}\nolimits} \right)\) and \(U\left( {\mathop{\rm v}\nolimits} \right)\)].

A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.

37. Construct a random \({\bf{4}} \times {\bf{4}}\) matrix Aand test whether \(\left( {A + I} \right)\left( {A - I} \right) = {A^2} - I\). The best way to do this is to compute \(\left( {A + I} \right)\left( {A - I} \right) - \left( {{A^2} - I} \right)\) and verify that this difference is the zero matrix. Do this for three random matrices. Then test \(\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^{\bf{2}}}\) the same way for three pairs of random \({\bf{4}} \times {\bf{4}}\) matrices. Report your conclusions.
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