Solving Simultaneous Equations (An introduction to Linear Algebra)

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					  Solving Simultaneous Equations
(An introduction to Linear Algebra)
           Systems of Equations
• Many geoscience problems require using systems
  of equations.
• A common application for systems of equations
  involves estimating the fraction of several
  components in a mixture.
• Examples:
  – Estimating the mineralogical composition of a rock
    based on chemical analyses
  – Deriving source rocks contribution to glacial till based
    on the chemistry of various size fractions.
          Model Compositions
• A calculated source composition of till or the
  mineralogical composition of a rock based on
  its major oxide chemistry is a modeled
              How does it work?
• Let’s assume that each element (or chemical
  species) exists in the same proportion in each
• The way each element is distributed can be
  expressed linearly:
                                 Where the coefficient a is the ij
a11x1+a12x2+…+a1mxm=b1           known content of element i in
                                 component j. b is the total
a21x1+a22x2+…+a2mxm=b2           content of element i in the
                                         For m unknown components,
an1x1+an2x2+…+anmxm=bn                   the proportions xj to be solved
                                         are the same for n elements.

What we know:
All we need is xm

If m=n then we have the same number of
equations as we have of unknowns! This is a
simple system of linear equations.
(from Mäkinen and Gustavsson, 1998)

• A stone count near glacial till in Finland
  roughly contains 20% granitoids, 53%
  quartzite, and 20% amphibolite.
• This does not agree with the chemical
  composition of till corresponding to these 3
  rock types.
• A layer of gravel sand is found beneath the till.
  Based on appearance, it is thought that the
  gravel-sand layer is where the till originated.
             Setting up the model
• Include the following chemical species which exist in
  known amounts in each of the components:
• Al2O3, CaO, MgO, Cr, Ni, Pb, Rb, Sr, V, Y, Na2O, SiO2,
  TiO2, K2O
• Have bulk analysis of the 4 components: granitoids
  (A), quartzite(B), amphibolite(C), gravel sand(D).
• Have bulk analysis of the till (T).
Let’s take the example of Al2O3:
       AAl2O3xA + BAl2O3xB+ CAl2O3xC + DAl2O3xD = TAl2O3
           ACaOxA + BCaOxB+ CCaOxC + DCaOxD = TCaO
          AK2OxA + BK2OxB+ CK2OxC + DK2OxD = TK2O
• Check to see that it
  adds up! xA = 0.26, xB =
  0.03, xC = 0.09, xD = 0.62
• Notice that the equations are linear, of the form Ax = B.
• Need to find an easy way to multiple equations
• We can represent the equations in the last example as
  coefficients and constants, omitting the variables.
AAl2O3+ BAl2O3+ CAl2O3 + DAl2O3 = TAl2O3   xA       TAl2O3
    ACaO + BCaO+ CCaO+ DCaO = TCaO         xB   =   TCaO
                  …                        xC       …
   AK2O + BK2O+ CK2O + DK2O = TK2O         xD       TK2O

              A                            x =      B
              Matrix Algebra
• Matrices provide a theoretical and practical
  way of solving problems.
• Excel is a natural medium by which to
  construct, analyze and solve matrices.
           What is a Matrix?

             Food   Utilities   Health
Family A     a      b           c
Family B     d      e           f
What is a Matrix?
Size of a Matrix
• Defined by number of
   ROWS and number of
• In this case, the matrix on
   the left has 2 rows and 3
   columns so it is a (2x3)
• More generally, it has m
   rows and n columns so we
   say it’s an (mxn) matrix.
The elements of a matrix can be variables or
functions instead. It provides a grid
arrangement for data, much like the way Excel
manages data.

Matrices can be added, subtracted, multiplied
and, in general, manipulated to extract
          Examples of matrices
• A matrix that has equal
  number of rows and
  columns, i.e., m=n is
  called a square matrix.
• A matrix with 4 rows
  and 2 columns is a 4x2
• How would you
  describe the matrix to
  the right in terms of
            Matrix Operations
• To add or subtract two
  matrices, simply add
  the elements together.
  Matrices must be the
  same size (i.e., same

• Or more generally:
           Matrix Operations
• As implied by the last example, you can apply
  an operation to an entire matrix. Adding a
  matrix to itself is the same as doubling a
  matrix. Therefore we can multiply a matrix by
  a constant.
           Matrix Operations
• What about multiplying two matrices?
• Can only be done if the number of rows in
  matrix A equals the number of columns in
  matrix B.
• For example:
    How to multiply matrices?


In other words if matrix A is an mxn matrix and
matrix B is a kxl matrix, n must equal k in order to
 multiply them. The result will be an mxl matrix.
However AB ≠ BA!
           More Properties:
    Adding and Multiplying Matrices
• If A, B, and C are matrices and a and b are
                  (A+B)C = AC + BC
                 A(B+C) = AB + AC

               A(A+B) = aA + bB
               (a+b)A = aA + bA
            The Identity Matrix
If we have the matrices

It’s easy to see that

            The Identity Matrix

We call these two forms linear combinations. From
 this we can see how a matrix can behave like the
 number 1 behaves in normal multiplication.
           The Identity Matrix
A matrix that behaves
like the number 1.

For any nxn matrix A,
AIn = InA = A
              Invertible Matrices
An nxn matrix A is invertible if AB = BA = In
Where In is the Identity Matrix. If this is the case, B is the
inverse of A. Check to see if this is true:

The notation for “inverse of matrix A” is A-1
For a matrix to be invertible, it must be square.
        Using Invertible Matrices
• How do we use them? Remember that we
  want to solve systems of equations of the
  form Ax = B
• Multiply each side by the inverse of A.
                 A-1(AX) = A-1(B)
• Regroup the factors
                  (A-1A)X = A-1B
• Regroup the factors, and we know that A-1A= I
                     X = A-1B
 3x + 2y – 5z = 12
 x – 3y + 2z = -13
 5x – y + 4z = 10

Coefficients matrix Variables matrix   Constants matrix
      3 2 -5               x                  12
      1 -3 2               y                 -13
      5 -1 4                z                 10
                Excel Commands
You will need to know
   a)How to transpose matrix A. Use the =TRANSPOSE(A)
   a)how to multiply matrix A-1 by matrix B – use the
      =MMULT(range of cells from matrix A-1, range of cells for
      matrix B) function. Make sure your matrices are
      compatible for multiplication.
   b) how to invert a matrix A; use the function =MINVERSE(A)

As with any Excel formula, you may combine various operations
in one cell.
Because you are selecting multiple cells for input and output,
make sure you hold the CTRL-SHIFT keys when pressing ENTER.