Determinants are mathematical objects which are very useful in the analysis and solution of systems of linear equations. As
shown in Cramer's Rule, a nonhomogeneous system of linear equations has a nontrivial solution Iff the
determinant of the system's Matrix is Nonzero (so that the Matrix is nonsingular). A
determinant is defined to be

(1) |

(2) |

(3) |

(4) |

Given an determinant, the additive inverse is

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

The determinant of a Matrix Transpose equals the determinant of the original Matrix,

(13) |

(14) |

Let be a small number. Then

(15) |

(16) |

Important properties of the determinant include the following.

- 1. Switching two rows or columns changes the sign.
- 2. Scalars can be factored out from rows and columns.
- 3. Multiples of rows and columns can be added together without changing the determinant's value.
- 4. Scalar multiplication of a row by a constant multiplies the determinant by .
- 5. A determinant with a row or column of zeros has value 0.
- 6. Any determinant with two rows or columns equal has value 0.

(17) |

For a Matrix, the determinant is

(18) |

(19) |

(20) |

(21) |

If is an Matrix with Real Numbers, then
has the interpretation as the oriented -dimensional Content of the Parallelepiped spanned
by the column vectors , ..., in . Here, ``oriented'' means that, up to a change of or
Sign, the number is the -dimensional Content, but the Sign depends on the ``orientation'' of
the column vectors involved. If they agree with the standard orientation, there is a Sign; if not, there is a
Sign. The Parallelepiped spanned by the -D vectors through is the collection
of points

(22) |

There are an infinite number of determinants with no 0 or entries having unity determinant. One
parametric family is

(23) |

(24) |

**References**

Arfken, G. ``Determinants.'' §4.1 in *Mathematical Methods for Physicists, 3rd ed.*
Orlando, FL: Academic Press, pp. 168-176, 1985.

Guy, R. K. ``Unsolved Problems Come of Age.'' *Amer. Math. Monthly* **96**, 903-909, 1989.

Guy, R. K. ``A Determinant of Value One.'' §F28 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 265-266, 1994.

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1999-05-24