Function DETERMINANT

• Defined in theory: Basic-matrix-algebra
• Source code: basic-matrix.lisp

Slots on this function:

Documentation:

Function that returns the determinant of a matrix It has relevant informations such as if determinant = 0 the the matrix cannot be inverted

Instance-Of: Function

Implication Axioms:

(=> (And (Square-Matrix ?M))
    (= (Determinant ?M)
       (Cond ((= 1 (Size ?M)) (Value ?M 1 1))
             ((< 1 (Size ?M))
              (Summation (Lambda (?J)
                                 (* (Value ?M 1 ?J)
                                    (Cofactor ?M 1 ?J)))
                         1
                         (Size ?M))))))

Other Related Axioms:

(=> (And (Square-Matrix ?M))
    (= (Determinant ?M)
       (Cond ((= 1 (Size ?M)) (Value ?M 1 1))
             ((< 1 (Size ?M))
              (Summation (Lambda (?J)
                                 (* (Value ?M 1 ?J)
                                    (Cofactor ?M 1 ?J)))
                         1
                         (Size ?M))))))

(=> (= (Cofactor ?M ?I ?J) ?Cof)
    (= ?Cof
       (* (Expt -1 (+ ?I ?J))
          (Determinant (Matrix-Less-Row-And-Column ?M ?I ?J)))))

(Instance-Of (Zero-Element (Determinant ?M)) Not)

(Not (Zero-Element (Determinant ?M)))

(<=> (Invertible-Matrix ?M)
     (And (Square-Matrix ?M) (Not (Zero-Element (Determinant ?M)))))