<#2889#>Chemical Engineering 535<#2889#>
<#2890#>Fall, 1996<#2890#>
<#2891#>B. Bernstein, Instructor<#2891#>
Assignment 9 

 
Read:Sections 7.1, 7.2, 7.3, 7.5, 7.6, 7.7, 7.8, 7.9, 
8.1
 
Problems:page 360 # 1, 5, 12;page 375 # 
1,3,11,27
 
		page 399 # 1, 3, 5, 13;,page 422 # 1, 5, 11, 13
 
		page 431 # 1, 2;page 463 # 1, 2, 3, 4

<#3206#>Fall, 1996<#2896#>Chemical Engineering 535, Assignment 9<#2896#><#2897#>page\ <#2897#><#3206#> <#2898#>Please Note: <#2898#> This assignment covers a lot of material on matrices. In order to keep it feasible, only relatively few homework examples have been assigned. Please choose extra ones to do for yourself wherever you feel that you need more practice. If you are weak on matrix addition, please do some problems on page 355.

A matrix is a rectangular array of numbers: an #tex2html_wrap_inline3253# matrix

#displaymath3223#

which has m rows and n columns. The #tex2html_wrap_inline3259# are called <#2914#>elements<#2914#> or <#2915#>components<#2915#> of <#2916#>A<#2916#>. The following are permissible matrix operations:

  1. an #tex2html_wrap_inline3261# matrix can be added <#2918#><#2918#> only to itself or to another #tex2html_wrap_inline3263# matrix: <#2919#>A+B=C<#2919#> says that #tex2html_wrap_inline3267#, #tex2html_wrap_inline3269#, #tex2html_wrap_inline3271#.
  2. If #tex2html_wrap_inline3273# is a scalar, then #tex2html_wrap_inline3275# means #tex2html_wrap_inline3277#, #tex2html_wrap_inline3279#, #tex2html_wrap_inline3281#
See Theorem 1, page 354, for properties of matrix addition. An #tex2html_wrap_inline3283# matrix is a <#2928#>column matrix<#2928#> or <#2929#>column vector.<#2929#> A #tex2html_wrap_inline3285# matrix is a <#2930#>row matrix<#2930#> or <#2931#> row vector.<#2931#> If

#displaymath3224#

Then the dot product of <#2937#>a<#2937#> and <#2938#>b<#2938#> is

#displaymath3225#

<#2941#>Matrix multiplication<#2941#> <#2942#>AB<#2942#> is possible only if the number of columns of <#2943#>A<#2943#> equals the number of rows of <#2944#> B<#2944#>: If <#2945#>A<#2945#> is an #tex2html_wrap_inline3287# matrix and <#2946#>B<#2946#> is a #tex2html_wrap_inline3289# matrix, then #tex2html_wrap_inline3291# is an #tex2html_wrap_inline3293# matrix for which

#displaymath3226#

is the dot product of row i of <#2953#>A<#2953#> with column j of <#2954#>B<#2954#>. The <#2955#>transpose,<#2955#> #tex2html_wrap_inline3299# of the #tex2html_wrap_inline3301# matrix <#2957#>A<#2957#> is the #tex2html_wrap_inline3303# matrix whose rows are the columns of <#2958#>A<#2958#> (or, equivalently, whose columns are the rows of <#2959#> A<#2959#>). The <#2960#>system of linear equations<#2960#>

#equation2961#

can be written in matrix form as

#displaymath3227#

<#2999#>A<#2999#> is called the <#3000#>coefficient matrix,<#3000#> <#3001#>x<#3001#> is the matrix of unknowns and <#3002#>b<#3002#> is the <#3003#>right hand side.<#3003#> The information in equation #eq1#3004> can be summarized in the <#3005#>augmented matrix,<#3005#> namely

#equation3006#

<#3021#>Gauss-Jordan elimination<#3021#> can be applied to the system of equations (#eq1#3022>), or, equivalently, to the augmented matrix, (#eq2#3023>) to find the solutions if they exist. The permitted operations (page 368 - elementary row operations) are as follows:

  1. Multiply (or divide) one row (equation) by a non-zero number (scalar)
  2. Add a multiple of one row (equation) to another
  3. Interchange two rows (equations)
There are two parts to the process; elimination and back-substitution. The objective of elimination is to change the augmented matrix into a form, #tex2html_wrap_inline3305# i.e.~to change the system of equations into one of the form:

#equation3027#

for which which #tex2html_wrap_inline3307# if i;SPMgt;j.

If
one or more of the equations then reads #tex2html_wrap_inline3311#, there are no solutions.
Elseif
There remain exactly the same number of non-zero equations, m, as unknowns, one may solve from the bottom up. This is known as <#3034#>back substitution<#3034#>.
Else
there are fewer non-zero equations than unknowns, say p of them, one may put n-p of the variables on the right hand side and, with arbitrary values of these unknowns, one may solve for the other p variables from the bottom up as before by back substitution. In this case, there are multiple solutions.
<#3036#>Square Matrices<#3036#> #tex2html_wrap_inline3317#.
  • Symmetric matrix: #tex2html_wrap_inline3319#
  • Skew-Symmetric matrix: #tex2html_wrap_inline3321#
  • Upper triangular matrix: #tex2html_wrap_inline3323# if i;SPMgt;j.
  • Lower triangular matrix: #tex2html_wrap_inline3327# if i;SPMlt;j.
  • Diagonal matrix: #tex2html_wrap_inline3331# if #tex2html_wrap_inline3333#.
  • Main Diagonal: The elements #tex2html_wrap_inline3335#.
  • Unit matrix: #tex2html_wrap_inline3337# is a diagonal matrix with #tex2html_wrap_inline3339#. Then #tex2html_wrap_inline3341# and #tex2html_wrap_inline3343# whenever the multiplication is possible.
Most, but not all #tex2html_wrap_inline3345# matrices <#3053#>A<#3053#> have an inverse #tex2html_wrap_inline3347#. If so, then <#3056#>A<#3056#> is unique and

#displaymath3228#

so that the solution to #tex2html_wrap_inline3349# is #tex2html_wrap_inline3351#. The inverse, or whether it exists, can be found, by Gauss-Jordan elimination elimination: Form the matrix

#displaymath3229#

and perform the Gauss Jordan eliminations steps until the unit matrix is on the left hand side: #tex2html_wrap_inline3353#. Then the matrix on the right hand side, <#3086#>C<#3086#>, will be the inverse matrix. <#3087#>Determinants<#3087#> To each <#3088#>square<#3088#> #tex2html_wrap_inline3355# matrix, <#3089#>A<#3089#>, there is associated a number, #tex2html_wrap_inline3357#, called its <#3091#>determinant.<#3091#> The determinant is evaluated as follows:

  • If n=1, then we may write #tex2html_wrap_inline3361#, where a is the only component of <#3094#>A<#3094#>. In that case, #tex2html_wrap_inline3365#.
  • If n;SPMgt;1, then one proceeds by minors as follows:
    • The minor, #tex2html_wrap_inline3369# of the component #tex2html_wrap_inline3371# is the determinant of the matrix which is left after striking out row i and column j of <#3099#>A<#3099#>
    • The cofactor, #tex2html_wrap_inline3377# of the component #tex2html_wrap_inline3379# is #tex2html_wrap_inline3381#.
    • Choose <#3104#>any<#3104#> row, i, or <#3105#>any<#3105#> column, j. Then

      #displaymath3230#

      The first sum is called the <#3113#>expansion in terms of row<#3113#> i. The second sum is called the <#3114#>expansion in terms of column<#3114#> j.

<#3117#>Properties of Determinants<#3117#> (See Section 7.8).
Property 1
If any row or column of <#3119#>A<#3119#> is zero, then #tex2html_wrap_inline3391#.
Property 2
If a single row (or single column) of <#3121#>A<#3121#> is multiplied by a scalar, c, then the determinant of the resulting matrix is #tex2html_wrap_inline3395#.
Property 3
If one row (or one column) of <#3123#>C<#3123#> is the sum of the corresponding row (or column) of <#3124#>A<#3124#> with that of <#3125#>B<#3125#>, all other elements of <#3126#>A<#3126#>, <#3127#>B<#3127#> and <#3128#>C<#3128#> being the same, then #tex2html_wrap_inline3397#. Please note that in general the determinant of a matrix sum of two matrices differs from the sum of their determinants.
Property 4
Interchanging two rows (or two columns) of <#3132#>A<#3132#> multiplies #tex2html_wrap_inline3399# by -1.
Property 5
If two rows of <#3134#>A<#3134#> are equal (or if two columns of <#3135#>A<#3135#> are equal), then #tex2html_wrap_inline3403#.
Property 6
If a row of <#3137#>A<#3137#> is a scalar multiple of another row of <#3138#>A<#3138#> (or if a column of <#3139#>A<#3139#> is a scalar multiple of another colum of <#3140#>A<#3140#>), then #tex2html_wrap_inline3405#.
Property 7
If a scalar multiple of one row of <#3142#>A<#3142#> is added to another of its rows (or if a scalar multiple of a column of <#3143#>A<#3143#> is added to another of its columns), then #tex2html_wrap_inline3407# remains unchanged.
There follow a few theorems relative to determinants:
Theorem 1
(page 417)

#displaymath3231#

Theorem 2
(page 417)

#displaymath3232#

Theorem 3
(page 419)

#displaymath3233#

The determinant of the matrix product is the product of the determinants.

<#3161#>Cramer's Rule<#3161#> provides, at least in theory, a method of solving systems of linear equations with the use of determinants, and is actually usable as such for small enough systems of equations. (Numerical methods prefer, for good reason, to be based on Gauss-Jordan elimination or related procedures. However, Cramer's rule is still important in understanding matrices.) We describe it as follows:

Consider the #tex2html_wrap_inline3409# system of equations represented in matrix form by #tex2html_wrap_inline3411#, where <#3165#>A<#3165#> is a given #tex2html_wrap_inline3413# matrix, <#3166#>b<#3166#> is a given #tex2html_wrap_inline3415# matrix and <#3167#>x<#3167#> is an #tex2html_wrap_inline3417# matrix of unknowns, #tex2html_wrap_inline3419#, #tex2html_wrap_inline3421#. Let #tex2html_wrap_inline3423# and let #tex2html_wrap_inline3425# be the determinant of the matrix formed by replacing column i of <#3169#>A<#3169#> by <#3170#>b<#3170#>. Then a sequence of algebraic operations, involving additions, subtractions and multiplications, <#3171#>but no divisions,<#3171#> will lead to

#displaymath3234#

It follows then that if D=0 and one of the #tex2html_wrap_inline3431#, then there are no solutions. On the other hand, if #tex2html_wrap_inline3433#, then the solution is unique and is given by

#displaymath3235#

Indeed, it also follows that if #tex2html_wrap_inline3435#, then #tex2html_wrap_inline3437# exists and is given by

#displaymath3236#

Indeed, if #tex2html_wrap_inline3439#, it can also be shown that #tex2html_wrap_inline3441# does not exist. The <#3183#>homogeneous system<#3183#> is #tex2html_wrap_inline3443#. If #tex2html_wrap_inline3445#, then the homogeneous system has only the unique solution #tex2html_wrap_inline3447# and the solution of the <#3190#>nonhomogeneous system,<#3190#> #tex2html_wrap_inline3449# is unique. Otherwise the homogeneous system has multiple solutions and the non-homogeneous system either has no solution or has multiple solutions. <#3194#>Eigenvalue Problems:<#3194#> Consider a square matrix <#3195#>A<#3195#>. The eigenvalue problem is to find values of a scalar, #tex2html_wrap_inline3451#, and corresponding <#3196#>non-trivial<#3196#> vectors, #tex2html_wrap_inline3453#, such that #tex2html_wrap_inline3455#. To do this, solve

#displaymath3237#

which is a polynomial equation of order n, the order of the matrix <#3204#>A<#3204#>. For each root, there is at least one eigenvector. Eigenvectors are determined only to within an arbitrary scalar multiple: Only directions, not sizes, of the vectors are determined. If all the n eigenvalues are distinct, then exactly n such directions are determined.