# DET

 Syntax DET [array] Location Math Package

The function DET returns the determinant of a square matrix, meaning that the array (or the part passed) must have two dimensions of equal size, otherwise DET breaks with error -15 (bad parameter).

The array needs to be a floating point array, any other type (including integer arrays) will also produce error -15.

If no parameter is given, DET will use the array that has been supplied to the previously executed MATINV command as its source. If however, this command has not yet been used, DET without a parameter will stop with the error -7 (not found).

You may ask what a determinant is? Briefly speaking, it represents a square matrix by a single number so that important characteristics of the matrix can be deduced from it, eg. the matrix cannot be inverted if the determinant is zero.

Example

We will try to approach the eigenvalues of a matrix and list them all (the so-called “spectrum” of a matrix). Due to approximation errors and the simple algorithm employed, there can be more output values than there should be. This can be improved by increasing estep in line 130, but at the cost of speed.

The range of expected eigenvalues (eval1 to eval2) is adapted to the chosen matrix whose random elements only range between 0 and 1. There is no limit for the positive size n of the matrix, n=0 is allowed but does not make sense because CHARPOLY becomes constant:

```100 CLEAR: RANDOMISE 10: PRINT "Eigenvalues:"
110 n=2: DIM matrix(n,n), one(n,n)
120 MATRND matrix: MATIDN one
130 :
140 eval1=-1: eval2=1: esteps=200
150 eprec<(eval2-eval1)/estep)
160 c1=CHARPOLY(matrix,eval1): count%=0
170 FOR eval=eval1+eprec TO eval2 STEP eprec
180 c2=CHARPOLY(matrix,eval)
190 IF SGN(c1)<>SGN(c2) THEN PRINT eval
200 c1=c2: count%=count%+1
210 AT#0,0,0: PRINT#0,INT(100*count%/esteps);"%"
220 END FOR eval
230 PRINT "absolute fault:"!eprec
240 :
250 DEFine FuNction CHARPOLY(matrix,lambda)
260   LOCal diff(n,n),i
270   FOR i=1 TO n: one(i,i)=lambda
280   MATSUB diff,matrix,one
290   RETurn DET(diff)
300 END DEFine CHARPOLY
```

In practice, a Newton iteration algorithm (or better) would be used.

CROSS-REFERENCE

MATINV co-operates closely with DET, so that for each of them a matrix parameter can be omitted if the other function has been called before; MATINV calls DET internally. In the example, we used the MATRND, MATIDN, SGN and MATSUB keywords which are all part of the same Toolkit.