# Lesson 09: Matrix Algebra

## Objectives

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

## 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\$