## DETERMINANTS AND INVERSES

Throughout this section, all the matrices are square and n X n.

Before we give a formal definition of the determinant of a square matrix, let us give some examples. The determinant of a 1 X 1 matrix, or a scalar, is the scalar itself. Consider a 2 X 2 matrix

д __ an an

Cl 2i #22

Its determinant, denoted by |A| or det A, is defined by (11.3.1) |A| = <211^22 — <221«12-

The determinant of а З X 3 matrix

ап |
а12 |
аЪ |

а21 |
а 22 |
а23 |

а31 |
а32 |
аЪЪ |

is given by

= «цяггя’зз — «11«32«23 — «21«12«33 + «21 «32 «13 + «31 «12«23 — «31«22«13 •

Now we present a formal definition, given inductively on the assumption that the determinant of an (n – 1) X (n — 1) matrix has already been defined.

DEFINITION 11...

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