# 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
composition.
How does it work?
• Let’s assume that each element (or chemical
species) exists in the same proportion in each
component.
• 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
mixture.
…
For m unknown components,
an1x1+an2x2+…+anmxm=bn                   the proportions xj to be solved
are the same for n elements.
an1x1+an2x2+…+anmxm=bn

What we know:
a
b
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.
Example
(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
Solution
• 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
simultaneously.
• 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?

Monthly
Expenses
Food   Utilities   Health
Family A     a      b           c
Family B     d      e           f
What is a Matrix?
Matrices
Size of a Matrix
• Defined by number of
ROWS and number of
COLUMNS.
• In this case, the matrix on
the left has 2 rows and 3
columns so it is a (2x3)
matrix.
• More generally, it has m
rows and n columns so we
say it’s an (mxn) matrix.
Matrices
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
information.
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
matrix.
• How would you
describe the matrix to
the right in terms of
mxn?
Matrix Operations
• To add or subtract two
the elements together.
Matrices must be the
same size (i.e., same
mxn)

• 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?

Or

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:
• If A, B, and C are matrices and a and b are
numbers,
(A+B)C = AC + BC
A(B+C) = AB + AC

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

It’s easy to see that

And
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
Example
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.

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