## Matrices and Determinoids:, Volume 1Originally published in 1913, this book forms part of a three-volume work created to expand upon the content of a series of lectures delivered at the University of Calcutta during the winter of 1909-10. The chief feature of all three volumes is that they deal with rectangular matrices and determinoids as distinguished from square matrices and determinants, the determinoid of a rectangular matrix being related to it in the same way as a determinant is related to a square matrix. An attempt is made to set forth a complete and consistent theory or calculus of rectangular matrices and determinoids. The first volume contains the most fundamental portions of the theory and concludes with the solution of any system of linear algebraic equations, which is treated as a special case of the solution of a matrix equation of the first degree. |

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### Contents

CHAPTER | 1 |

Determinoid deﬁned as an algebraical sum of complete derived | 12 |

Some simple properties of determinoids 1921 | 19 |

The sign of a derived product is independent of the order of arrange | 47 |

moves affects of derived sequences | 55 |

CHAPTER IV | 86 |

Theorems concerning the affects of complementary corranged derived | 102 |

Reciprocal matrices and reciprocal determinoids 110111 | 110 |

Product of three matrices taken in prescribed order reduction | 184 |

Special cases of a standard product formed by a chain of matrix | 207 |

one factor a unit matrix one factor a scalar matrix | 219 |

activities are not equal 232244 | 243 |

of a product | 253 |

Reciprocal and conjugate reciprocal of a standard product of square | 261 |

RANK OF A MATRIX AND CONNEGHONS BETWEEN | 265 |

Changes in the value of a determinoid due to inversions in the orders | 286 |

Expansion of a determinoid in terms of the simple minor determinants | 116 |

Algebraical sum of the products obtained from two ﬁxed complementary | 138 |

CHAPTER VI | 153 |

Properties of the passive rows in a product of two matrices partial | 164 |

equivalence of con | 173 |

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### Common terms and phrases

a1 bl active rows aﬂ'ect algebraical sum augmented matrix co-factor complete derived product conjugate reciprocal connected corresponding passive rows deduced deﬁned deﬁnition denoted derived sequence determinants of order equivalent ﬁnal passive rows ﬁnite solution ﬁrst factor matrix ﬁxed formula forward moves fundamental matrix fundamental sequence given equation given matrices horizontal affect horizontal and vertical identical inﬁnite solutions inner elements integers interchanged rows inverse matrix ith horizontal row ith vertical jth vertical row long rows matrix equation minor matrix minor of order multiplied non-vanishing derived determinant notation number of inversions obtained postﬁxing preﬁxing product matrix proved rank relative affect relative aﬁect relative order replaced respectively result rows of 0’s rule of signs satisﬁed scalar matrix short rows sign determined similar matrices simple minor determinants simple minor determinoid simple minor matrices square matrix sub-article suﬂixes total number unaltered unaugmented unconnected undegenerate vanish vertical affect zero zero matrix