Lesson 09: Matrix Algebra

Objectives

  • Learn the basic operations of the matrix algebra.
  • Learn how to solve simultaneous equations by using MATLAB matrix operations.

 

Background

 

 

 

Questions

Dot Product

  1. Compute the dot product of the following pairs of vectors, and then show that \(A \cdot B = B \cdot A\)
    1. \(A = \left[ {\begin{array}{*{20}{c}}1&3&5\end{array}} \right]\)  \(B = \left[ {\begin{array}{*{20}{c}}{ - 3}&{ - 2}&4\end{array}} \right]\)
    2. \(A = \left[ {\begin{array}{*{20}{c}}0&{ - 1}&{ - 4}&{ - 8}\end{array}} \right]\)  \(B = \left[ {\begin{array}{*{20}{c}}4&{ - 2}&{ - 3}&{24}\end{array}} \right]\)
  2. Use a dot product and the shopping list in the following Table to determine your total bill at the grocery store.
    ItemNumber NeededCost
    Milk 2 gallons \(\$ 3.50\) per gallon
    Eggs 1 dozen \(\$ 1.25\) per dozen
    Cereal 2 boxes \(\$ 4.25\) per box
    Soup 5 cans \(\$ 1.55\) per can
    Cookies 1 package \(\$ 3.15\) per package
  3. Bomb calorimeters are used to determine the energy released during chemical reactions. The total heat capacity of a bomb calorimeter is defined as the sum of the products of the mass of each component and the specific heat capacity of each component, or
    \[CP = \sum\limits_{i = 1}^n {{m_i}{C_i}} \]
    where
         \({{m_i}} = \)mass of component i, g
         \({{C_i}} = \) heat capacity of component, i , J/(gK)
         \(CP = \) total heat capacity, J/K
    Find the total heat capacity of a bomb calorimeter, using the thermal data in the following Table.
    ComponentMass, gHeat Capacity, J?(gK)
    Steel 250 0.45
    Water 100 4.2
    Aluminum 10 0.90

Matrix Multiplication

  1. Compute the matrix product \(A*B\) of the following pairs of matrices:
    1. \(A = \left[ {\begin{array}{*{20}{c}}{12}&4\\3&{ - 5}\end{array}} \right]\)  \(B = \left[ {\begin{array}{*{20}{c}}2&{12}\\0&0\end{array}} \right]\)
    2. \(A = \left[ {\begin{array}{*{20}{c}}1&3&5\\2&4&6\end{array}} \right]\)  \(B = \left[ {\begin{array}{*{20}{c}}{ - 2}&4\\3&8\\{12}&{ - 2}\end{array}} \right]\)
    Show that \(A*B\) is not the same as \(B*A\).

Determinants and Inverses

  1. Recall that not all matrices have an inverse. A matrix is singular (i.e., it doesn't have an inverse) if its determinant equals 0 (i.e., \(\left| A \right| = 0\)). Use the determinant function to test whether each of the following matrices has an inverse:
    \(A = \left[ {\begin{array}{*{20}{c}}2&{ - 1}\\2&5\end{array}} \right]\), \(B = \left[ {\begin{array}{*{20}{c}}4&2\\2&1\end{array}} \right]\), \(C = \left[ {\begin{array}{*{20}{c}}2&0&0\\1&2&2\\5&{ - 4}&0\end{array}} \right]\)
    If an inverse exists, compute it.

Solving Linear Systems of Equations

  1. Solve the following system of equations, using both matrix left division and the inverse matrix method:
    1. \( - 2x + y = 3\)
      \( x+y=10\)
    2. \( 5x+3y-z=10\)
      \( 3x+2y+z=4\)
      \( 4x-y+3z=12\)
    3. \( 3x+y+z+w=24\)
      \( x-3y+7z+w=12\)
      \( 2x+2y-3z+4w=17\)
      \( x+y+z+w=0\)

 

 

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