Page 62 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 3

A-I AX = A-~B
IX = A-~B
x = A-~B
By postmultiplying the inverted matrix A-l by the matrix B, the unknown ma-
trix, X, is solved,

A-’ x B = X
-::;I
[ 4:; [ lE] = [:I

The column vector contains the unknown coefficients which we find to be equal to
x1 = 2 and x2 = 3. You will recall that it was stated that these were the coefficients
originally in the equation set, so we have recovered the proper values.
As an additional example of the solution of simultaneous equations by matrix
inversion, we can set the equations below into matrix form and solve for x1 and x2
by inversion,
2x1 +x2 = 4
3x1 4- 4x2 = 1

The steps in the inversion process can be written out briefly:

[; :]x[::]=[:]

415 -115 1
2. “ 34 -315 215

Therefore, the unknown coefficients are XI = 3 and x2 = -2.
It may be noted that the procedure just described is almost exactly the same
as the classical algebraic method of solving two simultaneous equations. In fact,
the solution of simultaneous equations is probably the most important applica-
tion of matrix inversion. The advantage of matrix manipulation over the “try it
and see” approach of ordinary algebra is that it is more amenable to computer
programming. Almost all of the techniques described in subsequent chapters of
this book involve the solution of sets of simultaneous equations. These can be
expressed conveniently in the form of matrix equations and solved in the manner
just described.
Matrix inversion can, of course, be applied to square matrices of any size, and
not just the 2 x 2 examples we have investigated so far. Demonstrate this to yourself
by inverting the 3 x 3 matrix below:

If we need the inverse of a diagonal matrix, the problem is much simpler.
The inverse of a diagonal matrix is simply another diagonal matrix whose nonzero

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