Page 60 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 3

Inversion and Solution of Simultaneous Equations

Division of one matrix by another, in the sense of ordinary algebraic division, cannot
be performed. However, by utilizing the rules of matrix multiplication, an operation
can be performed that is equivalent to solving the equation
AxX=B

for the unknown matrix, X, when the elements of A and B are known. This is one of
the most important techniques in matrix algebra, and it is essential for the solution
of simultaneous equations such as those of trend-surface analysis and discriminant
functions. The techniques of matrix inversion will be encountered again and again
in the next chapters of this book.
The equation given above is solved by finding the inverse of matrix A. The
inverse matrix (or reciprocal matrix) A-l is one that satisfies the relationship A x
A-l = I. If both sides of a matrix equation are multiplied by A-l, the matrix A
is effectively removed from the left side. At the same time, B is converted into a
quantity that is the value of the unknown matrix X. The matrix A must be a square
matrix. Beginning with
AxX=B
premultiply both sides by the inverse of A, or A-l:
A-'xAxX=A-l xB

Since A-l x A = I and I x X = X, the equation reduces to

X = A-' X B (3.2)

Thus, the problem of division by a matrix reduces to one of finding a matrix that
satisfies the reciprocal relationship. In some situations, an inverse cannot be found
because division by zero is encountered during the inversion process. A matrix with
no inverse is called a singular matrix, and presents problems beyond the scope of
this chapter.
The inversion procedure may be illustrated by solving the following pair of
simultaneous equations in matrix form. The unknown coefficients are x1 = 2 and
x2 = 3. We will attempt to recover them by a process of matrix inversion and
multiplication:
4x1 + 10x2 = 38
10x1 + 30x2 = 110

This is a set of equations of the general type
AX=B

where A is a matrix of coefficients, X is a column vector of unknowns, and B is a
column vector of right-hand sides of the equations. In the specific set of equations
[ 1;: ;:] [;:I = [ 1;:]
given above, we have

To solve the equation, the matrix A will be inverted and B will be multiplied by A-l
to give the solution for X.

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