Page 61 - Statistics and Data Analysis in Geology

Matrix Algebra

It may not be apparent why the set of simultaneous equations can be set into the
matrix form shown. You can satisfy yourself on this point, however, by multiplying
the two terms, AX, to obtain the left-hand side of the simultaneous equation set:

Working through this multiplication, you will see that all of the terms are as-
sociated with the proper coefficients. By the rules of matrix multiplication,

Then, multiplying the bottom row,

We will solve the simultaneous equation set by first inverting the term A. Place the
A matrix beside an identity matrix, I, and perform all operations simultaneously on
both matrices. The purpose of each operation is to convert the diagonal elements
of A to ones and the off-diagonal elements to zeros. This is done by dividing rows
of the matrix by constants and subtracting (or adding) rows of the matrix from
other rows:

[ i y ] The matrix A is placed beside an identity matrix, I;
1. [ 1:

025
row one is divided by 4, the first element in the row, to
*. [li %] [ 0 11 produce 1 at all;

10 times row one is subtracted from row two to reduce

4. [ i ";] [ 02' '1 row two is divided by 5 to give 1 at u22, and
-0.5 0.2

10 le5 -Oms
2.5 times row two is subtracted fromrow one to reduce
5- [ 0 11 [ -0.5 0.21 the final off-diagonal element to 0.
The matrix is now inverted. Work may be checked by multiplying the original matrix
A by the inverted matrix, A-l, which should yield the identity matrix

4 10

Because
A-1A = I
the following identities hold:

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