13.23: Chapter Exercises
- Page ID
- 135941
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise 1: Vector Basics
Create the vectors A = [2 -1 4] and B = [3 5 -2]. Then calculate A + B, A - B, 4A, dot(A,B), cross(A,B), norm(A), and the unit vector in the direction of A.
Exercise 2: Perpendicular Vectors
Use the dot product to determine whether the vectors [1 2 -1] and [3 -1 1] are perpendicular. Explain your answer in a short comment in your script.
Exercise 3: Special Matrices
Create the matrix M = [4 2 9; 1 5 3; 8 6 7]. Use MATLAB to find the transpose, diagonal values, trace, lower triangular part, and upper triangular part.
Exercise 4: Write Your Own Trace/Diagonal Function
Write a function named matrixTraceDiag.m. The function receives a square matrix and returns two outputs: the diagonal of the matrix and the sum of the elements on the diagonal. Display a proper error message if the matrix is not square.
Exercise 5: Matrix Addition and Multiplication
Create two matrices A and B of compatible sizes. Demonstrate A + B, A - B, A*B, and A.*B where possible. Add comments explaining which operations require matching dimensions and which operations require the matrix multiplication rule.
Exercise 6: Solve a 2 by 2 System
Solve the following system using the backslash operator:
\(3*x1 + 2*x2 = 12\)
-\(4*x1 + 2*x2 = 4\)
Then rewrite the equations as straight lines and plot them to verify the intersection point.
Exercise 7: Change the System
Change the second equation in Exercise 6 to -6*x1 - 4*x2 = -24. Solve the new system and plot the two lines. How many solutions does the system appear to have?
Exercise 8: Four Unknowns
Solve the following system using matrix operations:
\(4*x1 - x2 + 3*x4 = 10\)
\(-2*x1 + 3*x2 + x3 - 5*x4 = -3\)
\(x1 + x2 - x3 + 2*x4 = 2\)
\(3*x1 + 2*x2 - 4*x3 = 4\)
Remember to include zero coefficients for variables that are missing from an equation.
Exercise 9: Engineering Application
A simple force balance produces the following equations for three unknown forces F1, F2, and F3:
\(F1 + 2*F2 - F3 = 10\)
\(2*F1 - F2 + 3*F3 = 25\)
\(-F1 + F2 + F3 = 5\)
Solve for the three forces using MATLAB. Then verify your answer by computing A*F.