Topic       : The GFA-Basic Compendium
Author      : GFA Systemtechnik GmbH
Version     : GFABasic.HYP v2.98 (12/31/2023)
Subject     : Documentation/Programming
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View Ref-FileMAT ADD a()=b()+c()
MAT ADD a(),b()
MAT ADD a(),x
MAT SUB a()=b()-c()
MAT SUB a(),b()
MAT SUB a(),x
MAT MUL a()=b()*c()
MAT MUL x=a()*b()
MAT MUL x=a()*b()*c()
MAT MUL a(),x
MAT NORM a(),0
MAT NORM a(),1
MAT DET x=a([i,j])[,n]
MAT QDET x=a([i,j])[,n]
MAT RANG x=a([i,j])[,n]
MAT INV a()=b()

a, b, c: names of numerical floating point fields
x: aexp (scalar value)
i, j, n: aexp

MAT ADD a()=b()+c() is only defined for matrix (vectors) of the same order,
e.g. DIM a(n,m),b(m,m),c(n,m) or DIM a(n),b(n),c(n). Array c is added to matrix
b, element by element, and the result is written to matrix a.

Example:

    DIM a(3,5),b(3,5),c(3,5)
    MAT SET b()=3
    MAT SET c()=4
    MAT PRINT b()
    PRINT STRING$(10,"-")
    MAT PRINT c()
    PRINT STRING$(10,"-")
    MAT ADD a()=b()+c()
    MAT PRINT a()

--> Outputs: 3,3,3,3,3
             3,3,3,3,3
             3,3,3,3,3
             ---------
             4,4,4,4,4
             4,4,4,4,4
             4,4,4,4,4
             ---------
             7,7,7,7,7
             7,7,7,7,7
             7,7,7,7,7

MAT ADD a(),b() is only defined for matrix (vectors) of the same order, e.g.
DIM a(n,m),b(n,m) or DIM a(n),b(n). Array b is added to matrix a, element by
element, and the result is written to matrix a. The original matrix a is lost
in the process.

Example:

    DIM a(3,5),b(3,5)
    MAT SET a()=1
    MAT SET b()=3
    MAT PRINT a()
    PRINT STRING$(10,"-")
    MAT PRINT b()
    PRINT STRING$(10,"-")
    MAT ADD a(),b()
    MAT PRINT a()

--> Outputs: 1,1,1,1,1
             1,1,1,1,1
             1,1,1,1,1
             ---------
             3,3,3,3,3
             3,3,3,3,3
             3,3,3,3,3
             ---------
             4,4,4,4,4
             4,4,4,4,4
             4,4,4,4,4

MAT ADD a(),x is defined for all matrix (vectors). Here, the scalar x is added
to matrix a, element by element, and the result is written to matrix a. The
original matrix a is lost in the process.

Example:

    DIM a(3,5)
    MAT SET a()=1
    MAT PRINT a()
    PRINT STRING$(10,"-")
    MAT ADD A(),5
    MAT PRINT a()

--> Outputs: 1,1,1,1,1
             1,1,1,1,1
             1,1,1,1,1
             ---------
             6,6,6,6,6
             6,6,6,6,6
             6,6,6,6,6

MAT SUB a()=b()-c() is only defined for matrix (vectors) of the same order,
e.g. DIM a(n,m),b(n,m),c(n,m) or DIM a(n),b(n),c(n). Array c is subtracted from
matrix b, element by element, and the result is written to matrix a.

Example:

    DIM a(3,5),b(3,5),c(3,5)
    MAT SET b()=5
    MAT SET c()=3
    MAT PRINT b()
    PRINT STRING$(10,"-")
    MAT PRINT c()
    PRINT STRING$(10,"-")
    MAT SUB a()=b()-c()
    MAT PRINT a()

--> Outputs: 5,5,5,5,5
             5,5,5,5,5
             5,5,5,5,5
             ---------
             3,3,3,3,3
             3,3,3,3,3
             3,3,3,3,3
             ---------
             2,2,2,7,2
             2,2,2,2,2
             2,2,2,2,2

MAT SUB a(),b() is only defined for matrix (vectors) of the same order, e.g.
DIM a(n,m),b(n,m) or DIM a(n),b(n). Array b is subtracted from matrix a,
element by element, and the result written to matrix a. The original matrix a
is lost in the process.

Example:

    DIM a(3,5),b(3,5)
    MAT SET a()=3
    MAT SET b()=1
    MAT PRINT a()
    PRINT STRING$(10,"-")
    MAT PRINT b()
    PRINT STRING$(10,"-")
    MAT SUB a(),b()
    MAT PRINT a()

--> Outputs: 3,3,3,3,3
             3,3,3,3,3
             3,3,3,3,3
             ---------
             1,1,1,1,1
             1,1,1,1,1
             1,1,1,1,1
             ---------
             2,2,2,2,2
             2,2,2,2,2
             2,2,2,2,2

MAT SUB a(),x is defined for all matrix (vectors). Here, the scalar x is
subtracted from matrix x, element by element, and the result is written to
matrix a. The original matrix a is lost in the process.

Example:

    DIM a(3,5)
    MAT SET a()=6
    MAT PRINT a()
    PRINT STRING$(10,"-")
    MAT SUB A(),5
    MAT PRINT a()

--> Outputs: 6,6,6,6,6
             6,6,6,6,6
             6,6,6,6,6
             ---------
             1,1,1,1,1
             1,1,1,1,1
             1,1,1,1,1

MAT MUL a()=b()*c() is defined for matrix of an "appropriate" order.  Arrays b
and c are multiplied with each other. The result of this multiplication is
written to matrix a. In order for the result to be defined, the matrix on the
left (matrix b in this case) must have the same number of columns as the matrix
on the right (c in this case) has rows.  Array a, in this case, must have as
many rows as b and as many columns as c, for example:

    DIM a(2,2),b(2,3),c(3,2)

Arrays are multiplied as "row by column", i.e. element a(i,j) is obtained by
multiplying the elements in the ith row of matrix b with the elements in the
jth column of matrix c, element by element, and then adding up the individual
products.

Example:

    DIM a(2,2),b(2,3),c(3,2)
    MAT SET b()=1
    DATA 1,2,-3,4,5,-1
    MAT READ c()
    MAT PRINT b(),5,1
    PRINT STRING$(18,"-")
    MAT PRINT c(),5,1
    PRINT STRING$(18,"-")
    MAT MUL a()=b()*c()
    MAT PRINT a(),5,1

--> Outputs:  1.0,  1.0,  1.0
              1.0,  1.0,  1.0
             ------------------
              1.0,  2.0
             -3.0,  4.0
              5.0, -1.0
             ------------------
              3.0,  5.0
              3.0,  5.0

With vectors instead of matrix, MAT MUL a()=b()*c() results in the dyadic (or
external) product of two vectors.

Example:

    DIM a(3,3),b(3),c(3)
    DATA 1,2,-3,4,5,-1
    MAT READ b()
    MAT READ c()
    MAT PRINT b(),5,1
    PRINT STRING$(18,"-")
    MAT PRINT c(),5,1
    PRINT STRING$(18,"-")
    MAT MUL a()=b()*c()
    MAT PRINT a(),5,1

--> Outputs:   1.0,   2.0, -3.0
             ------------------
               4.0,   5.0, -1.0
             ------------------
               4.0,   5.0, -1.0
               5.0,  10.0, -2.0
             -12.0, -15.0,  3.0

MAT MUL x=a()*b() is only defined for vectors with an equal number of elements.
The result x is the scalar product (the so-called interior product) of vectors
a and b. The scalar product of two vectors is defined as the sum of n products
a(i)*b(i),i=1,...,n.

Example:

    DIM b(3),c(3)
    DATA 1,2,-3,4,5,-1
    MAT READ b()
    MAT READ c()
    MAT PRINT b(),5,1
    PRINT STRING$(18,"-")
    MAT PRINT c(),5,1
    PRINT STRING$(18,"-")
    MAT MUL x=b()*c()
    PRINT x

--> Outputs: 1.0, 2.0, -3.0
             ------------------
             4.0, 5.0, -1.0
             ------------------
             17.0

MAT MUL x=a()*b()*c() is defined for qualified Vectors a and c as well as
qualified Matrix b().

Example:

    DIM a(2).b(2,3),c(3)
    DATA 1,2,-3,4,5
    MAT READ a()
    MAT READ c()
    MAT SET b()=1
    MAT PRINT a(),5,1
    PRINT STRING$(18,"-")
    MAT PRINT b(),5,1
    PRINT STRING$(18,"-")
    MAT PRINT c(),5,1
    PRINT STRING$(18,"-")
    MAT MUL x=a()*b()*c()
    PRINT x

--> Outputs:  1.0,  2.0
             ------------------
              1.0,  1.0,  1.0
              1.0,  1.0,  1.0
              1.0,  1.0,  1.0
             ------------------
             -3.0,  4.0,  5.0
             ------------------
             18.0

MAT NORM a(),0 or MAT NORM a(),1 are defined for matrix and vectors. MAT NORM
a(),0 normalises a matrix (a vector) by rows, MAT NORM a(),1 by columns. This
means that after a normalisation by rows (by columns) the sum of the squares of
all elements in each row (column) is identical at 1.

Example:

    DIM a(10,10),b(10,10),v(10)
    DATA 1,2,3,4,5,6,7,8,9,-1
    DATA 3.2,4,-5,2.4,5.1,6.2,7.2,8.1,6,-5
    DATA -2,-5,-6,-1.2,-1.5,-6.7,4.5,8.1,3.4,10
    DATA 5,-2.3,4,5.6,12.2,18.2,14.1,16,-21,-13
    DATA 4.1,5.2,16.7,18.4,19.1,20.2,13.6,14.8,19.4,18.6
    DATA 15.2,-1.8,13.6,-4.9,5.4,19.8,16.4,-20.9,21.4,13.8
    DATA -3.6,6,-8.2,-9.1,4,-2.5,2,3.4,6.7,8.4
    DATA 4.7,8.3,9.4,10.5,11,19,15.4,18.9,-20,12.6
    DATA 5.3,-4.7,6.1,6.5,6.9,-9.2,-10.8,4.3,5.6,9.1
    DATA 21.4,19.5,28.4,19.3,24.6,14.9,71.3,23.5,14.5,-12.3
    '
    CLS
    MAT READ a()
    MAT CPY b()=a() !Source matrix stored
    PRINT "Source matrix"
    PRINT
    MAT PRINT a(),7,2
    ~INP(2)
    CLS
    MAT NORM a(),0
    PRINT
    PRINT "Row:"
    PRINT
    MAT PRINT a(),7,2
    ~INP(2)
    '
    PRINT
    PRINT "Test:"
    PRINT
    FOR i%=1 TO 10
      MAT XCPY v()=a(i%,1)
      MAT MUL x=v()*v()
      PRINT x'
    NEXT i%
    PRINT
    ~INP(2)
    '
    CLS
    MAT CPY a()=b()
    MAT NORM a(),1
    PRINT "Column:"
    PRINT
    MAT PRINT a(),7,2
    ~INP(2)
    '
    PRINT
    PRINT "Test:"
    PRINT
    FOR i%=1 TO 10
      MAT CPY v()=a(1,i%) !Copy column a() in the vector v()
      MAT MUL x=v()*v()
      PRINT x'
    NEXT i%
    ~INP(2)

--> Outputs:
    Source matrix

     1.00,  2.00,  3.00,  4.00,  5.00,  6.00,   7.00,   8.00,   9.00,  -1.00
     3.20,  4.00, -5.00,  2.40,  5.10,  6.20,   7.20,   8.10,   6.00,  -5.00
    -2.00, -5.00, -6.00, -1.20, -1.50, -6.70,   4 50,   8.10,   3.40,  10.00
     5.00, -2.30,  4.00,  5.60, 12.20, 18.20,  14.10,  16.00, -21.00, -13.00
     4.10,  5.20, 16.70, 18.40, 19.10, 20.20,  13.60,  14.80,  19.40,  18.60
    15.20, -1.80, 13.60, -4.90,  5.40, 19.80,  16.40, -20.90,  21.40,  13.80
    -3.60,  6.00, -8.20, -9.10,  4.00, -2.50,   2.00,   3.40,   6.70,   8.40
     4.70,  8.30,  9.40, 10.50, 11.00, 19.00,  15.40,  18.90, -20.00,  12.60
     5.30, -4.70,  6.10,  6.50,  6.90, -9.20, -10.80,   4.30,   5.60,   9.10
    21.40, 19.50, 28.40, 19.30, 24.60, 14.90,  71.30,  23.50,  14.50, -12.30

    Row:

     0.06,  0.12,  0.18,  0.24,  0.30,  0.35,  0.41,  0.47,  0.53, -0.06
     0.18,  0.23, -0.29,  0.14,  0.29,  0.36,  0.42,  0.47,  0.35, -0.29
    -0.11, -0.28, -0.34, -0.07, -0.09, -0.38,  0.26,  0.46,  0.19,  0.57
     0.12, -0.06,  0.10,  0.14,  0.30,  0.45,  0.35,  0.40, -0.52, -0.32
     0.08,  0.10,  0.33,  0.36,  0.38,  0.40,  0.27,  0.29,  0.38,  0.37
     0.32, -0.04,  0.29, -0.10,  0.11,  0.42,  0.35, -0.44,  0.45,  0.29
    -0.19,  0.32, -0.44, -0.48,  0.21, -0.13,  0.11,  0.18,  0.36,  0.45
     0.11,  0.19,  0.21,  0.24,  0.25,  0.43,  0.35,  0.43, -0.46,  0.29
     0.23, -0.21,  0.27,  0.29,  0.31, -0.41, -0.48,  0.19,  0.25,  0.40
     0.23,  0.21,  0.30,  0.21,  0.26,  0.16,  0.76,  0.25,  0.15, -0.13

    Test:

    1 1 1 1 1 1 1 1 1 1

    Columns:

     0.04,  0.08,  0.08,  0.12,  0.13,  0.14,  0.09,  0.18,  0.20, -0.03
     0.11,  0.16, -0.13,  0.07,  0.14,  0.14,  0.09,  0.18,  0.13, -0.14
    -0.07, -0.21, -0.15, -0.04, -0.04, -0.15,  0.06,  0.18,  0.07,  0.28
     0.18, -0.09,  0.10,  0.17,  0.33,  0.41,  0.18,  0.35, -0.46, -0.36
     0.14,  0.21,  0.42,  0.57,  0.51,  0.46,  0.17,  0.33,  0.42,  0.52
     0.53, -0.07,  0.35, -0.15,  0.15,  0.45,  0.21, -0.46,  0.47,  0.38
    -0.13,  0.25, -0.21, -0.28,  0.11, -0.06,  0.03,  0.08,  0.15,  0.23
     0.17,  0.34,  0.24,  0.33,  0.30,  0.43,  0.20,  0.42, -0.44,  0.35
     0.19, -0.19,  0.15,  0.20,  0.19, -0.21, -0.14,  0.10,  0.12,  0.25
     0.75,  0.80,  0.72,  0.60,  0.66,  0.34,  0.90,  0.52,  0.32, -0.34

    Test:

    1 1 1 1 1 1 1 1 1 1

MAT DET x=a([i,j])[,n] calculates the determinants of a square matrix of the
(n,n) type. The row and column offsets are preset to a(0,0) or a(1,1),
depending on MAT BASE 0 or MAT BASE 1, assuming that OPTION BASE 1 is enabled.
It is also possible, however, to calculate the determinant of a square part
matrix. To do this, the row and column offsets of a() must be specified as i
and j, and the number of elements in the part matrix as n. A part matrix of the
(n,n) type is then created internally starting from the "position" ith row, jth
column.

Example:

    DIM a(10,10),b(4,4)
    DATA 1,2,3,4,5,6,7,8,9,-1
    DATA 3.2,4,-5,2.4,5.1,6.2,7.2,8.1,6,-5
    DATA -2,-5,-6,-1.2,-1.5,-6.7,4.5,8.1,3.4,10
    DATA 5,-2.3,4,5.6,12.2,18.2,14.1,16,-21,-13,3.8
    DATA 4.1,5.2,16.7,18.4,19.1,20.2,13.6,14.8,19.4,18.6
    DATA 15.2,-1.8,13.6,-4.9,5.4,19.8,16.4,-20.9,21.4,13.8
    DATA -3.6,6,-8.2,-9.1,4,-2.5,2,3.4,6.7,8.4,10.9
    DATA 4.7,8.3,9.4,10.5,11,19,15.4,18.9,-20,12.6
    DATA 5.3,-4.7,6.1,6.5,6.9,-9.2,-10.8,4.3,5.6,9.1
    DATA 21.4,19.5,28.4,19.3,24.6,14.9,71.3,23.5,14.5,-12.3
    '
    CLS
    MAT READ a()
    PRINT "Source matrix"
    PRINT
    MAT PRINT a(),7,2
    PRINT
    PRINT "Determinant: ";
    MAT DET x=a()
    PRINT x;
    MAT DET y=a(1,4),4
    PRINT
    PRINT "Determinant from a(1,4),4: ";
    PRINT y
    PRINT
    PRINT "Test: "
    PRINT
    MAT CPY b()=a(1,4),4,4 !copy to b()
    MAT PRINT b(),7,2
    MAT DET z=b() !Determinant from b()
    PRINT
    PRINT z

--> Outputs:
    Source Matrix

       1.00,  2.00,  3.00,  4.00,  5.00,  6.00,  7.00,  8.00,   9.00,  -1.00
       3.20,  4.00, -5.00,  2.40,  5.10,  6.20,  7.20,  8.10,   6.00,  -5.00
      -2.00, -5.00, -6.00, -1.20, -1.50, -6.70,  4.50,  8.10,   3.40,  10.00
       5.00, -2.30,  4.00,  5.60, 12.20, 18.20, 14.10, 16.00, -21.00, -13.00
       3.80,  4.10,  5.20, 16.70, 18.40, 19.10, 20.20, 13.60,  14.80,  19.40
      18.60, 15.20, -1.80, 13.60, -4.90,  5.40, 19.80, 16.40, -20.90,  21.40
      13.80, -3.60,  6.00, -8.20, -9.10,  4.00, -2.50,  2.00,   3.40,   6.70
       8.40, 10.90,  4.70,  8.30,  9.40, 10.50, 11.00, 19.00,  15.40,  18.90
     -20.00, 12.60,  5.30, -4.70,  6.10,  6.50,  6.90, -9.20, -10.80,   4.30
       5.60,  9.10, 21.40, 19.50, 28.40, 19.30, 24.60, 14.90,  71.30,  23.50

    Determinant: -2549840202186
    Determinant from a(1,4),4: -57.61200000001

    Test:
       4.00,   5.00,   6.00,   7.00
       2.40,   5.10,   6.20,   7.20
      -1.20,  -1.50,  -6.70,   4.50
       5.60,  12.20,  18.20,  14.10

    -57.61200000001

MAT QDET x=a([i,j])[,n] works in the same manner as MAT DET x=a([i,j])[,n],
except that it has been optimized for speed rather than accuracy. Both will
normally produce identical results. With "critical" matrix, whose determinant
is close to 0, you should always use MAT DET, though.

Example:

    DIM a(10,10)
    DATA 1,2,3,4,5,6,7,8,9,-1
    DATA 3.2,4,-5,2.4,5.1,6.2,7.2,8.1,6,-5
    DATA -2,-5,-6,-1.2,-1.5,-6.7,4.5,8.1,3.4,10
    DATA 5,-2.3,4,5.6,12.2,18.2,14.1,16,-21,-13,3.8
    DATA 4.1,5.2,16.7,18.4,19.1,20.2,13.6,14.8,19.4,18.6
    DATA 15.2,-1.8,13.6,-4.9,5.4,19.8,16.4,-20.9,21.4,13.8
    DATA -3.6,6,-8.2,-9.1,4,-2.5,2,3.4,6.7,8.4,10.9
    DATA 4.7,8.3,9.4,10.5,11,19,15.4,18.9,-20,12.6
    DATA 5.3,-4.7,6.1,6.5,6.9,-9.2,-10.8,4.3,5.6,9.1
    DATA 21.4,19.5,28.4,19.3,24.6,14.9,71.3,23.5,14.5,-12.3
    '
    CLS
    MAT READ a()
    PRINT "Source matrix"
    PRINT
    MAT PRINT a(),7,2
    PRINT
    PRINT "Determinant with MAT DET: ";
    MAT DET x=a()
    PRINT x;
    PRINT
    PRINT "Determinant with MAT QDET: ";
    MAT DET y=a()
    PRINT y;
    PRINT
    PRINT "Deviation: ";x-y

--> Outputs:
    Source Matrix

       1.00,  2.00,  3.00,  4.00,  5.00,  6.00,  7.00,  8.00,   9.00,  -1.00
       3.20,  4.00, -5.00,  2.40,  5.10,  6.20,  7.20,  8.10,   6.00,  -5.00
      -2.00, -5.00, -6.00, -1.20, -1.50, -6.70,  4.50,  8.10,   3.40,  10.00
       5.00, -2.30,  4.00,  5.60, 12.20, 18.20, 14.10, 16.00, -21.00, -13.00
       3.80,  4.10,  5.20, 16.70, 18.40, 19.10, 20.20, 13.60,  14.80,  19.40
      18.60, 15.20, -1.80, 13.60, -4.90,  5.40, 19.80, 16.40, -20.90,  21.40
      13.80, -3.60,  6.00, -8.20, -9.10,  4.00, -2.50,  2.00,   3.40,   6.70
       8.40, 10.90,  4.70,  8.30,  9.40, 10.50, 11.00, 19.00,  15.40,  18.90
     -20.00, 12.60,  5.30, -4.70,  6.10,  6.50,  6.90, -9.20, -10.80,   4.30
       5.60,  9.10, 21.40, 19.50, 28.40, 19.30, 24.60, 14.90,  71.30,  23.50

    Determinant with MAT DET: -2549840202186
    Determinant with MAT QDET: -2549840202186
    Deviation: 0

MAT RANG x=a([i,j])[,n] outputs the rank of a square matrix. As with MAT DET or
MAT QDET, you can select any row and column offset. The number of elements in
the part matrix must be specified with n. This creates a part matrix of the
(n,n) type internally, starting from the position ith row, jth column.

Example:

    DIM a(5,5)
    DATA 1,2,3,4,5
    DATA 3.2,4,-5,2.4,5.1
    DATA -2,4,-5,2.4,5.1
    DATA 5,-2.3,4,5.6,12.2
    DATA 4.1,5.2,16.7,18.4,19.1
    '
    CLS
    MAT READ a()
    PRINT "Source matrix"
    PRINT
    MAT PRINT a(),7,2
    PRINT
    PRINT "Rang from a(): ";
    MAT RANG x=a()
    PRINT x;
    PRINT
    PRINT "Rang from a(1,2),3: ";
    MAT RANG y=a(1,2),3
    PRINT y;
    PRINT

--> Outputs:
    Source matrix

       1.00,   2.00,   3.00,   4.00,   5.00
       3.20,   4.00,  -5.00,   2.40,   5.10
      -2.00,   4.00,  -5.00,   2.40,   5.10
       5.00,  -2.30,   4.00,   5.60,  12.20
       4.10,   5.20,  16.70,  18.40,  19.10

    Rang from a(): 5
    Rang from a(1,2),3: 2

MAT INV b()=a() is used to determine the inverses of a square matrix. The
inverse of matrix a() is written to matrix b(), hence b() must be of the same
type as a().

Example:

    DIM a(5,5),b(5,5),c(5,5)
    DATA 1,2,3,4,5
    DATA 3.2,4,-5,2.4,5.1
    DATA -2,4,-5,2.4,5.1
    DATA 5,-2.3,4,5.6,12.2
    DATA 4.1,5.2,16.7,18.4,19.1
    '
    CLS
    MAT READ a()
    PRINT "Source matrix a(): "
    PRINT
    MAT PRINT a(),7,2
    '
    MAT INV b()=a()
    PRINT
    PRINT "Inverse from a(): "
    PRINT
    MAT PRINT b(),7,2
    PRINT
    PRINT "Test b()*a()>Unity matrix ?"
    PRINT
    MAT MUL c()=b()*a()
    MAT PRINT c(),7,2

--> Outputs:
    Source matrix a():

     1.00,  2.00,  3.00,  4.00,  5.00
     3.20,  4.00, -5.00,  2.40,  5.10
    -2.00,  4.00, -5.00,  2.40,  5.10
     5.00, -2.30,  4.00,  5.60, 12.20
     4.10,  5.20, 16.70, 18.40, 19.10

    Inverse from a():

     0.00,  0.19, -0.19, -0.00, -0.00
     0.97,  0.02, -0.09, -0.10, -0.17
     0.71, -0.10, -0.10, -0.01, -0.12
    -1.65,  0.17,  0.11, -0.06,  0.39
     0.71, -0.12,  0.04,  0.09, -0.17

    Test b()*a()>Unity matrix ?

     1.00,  0.00,  0.00,  0.00,  0.00
     0.00,  1.00,  0.00,  0.00, -0.00
     0.00, -0.00,  1.00,  0.00, -0.00
    -0.00, -0.00, -0.00,  1.00,  0.00
    -0.00,  0.00,  0.00,  0.00,  1.00

Memo: MAT DET x=a(i,j),n   !causes the compiler to crash
      MAT DET x=a(i,j),2+6 !is also fatal
      MAT DET x=a(i,j),4   !seems to compile ok
      MAT DET x=a(i,j),n&  !seems to compile ok
      MAT QDET and MAT RANG seem to have the same problems.