Let us begin with the determinant of product of matrices. The proof of the property will be given here, after the rigorous definition of determinants.

Taking the absolute value, the property means that the ratio for the change of volume under the composition ** AB** is the product of the ratios under the transformations

Taking the signs, we may consider various combinations. For example, if 2 by 2 matrices ** A** and

(** u**,

so that ** AB** also preserves orientation (i.e.,

We also note that the product property implies

`det`(` A^{k}`) = (

The next property indicates that, as far as the determinant is concerned, there is no difference between rows and columns. In particular, one may use row operations as well as column operations. The proof of the property will be given here, after the rigorous definition of determinants.

The next property is quite useful for computation. The proof of the property will be given here, after the rigorous definition of determinants.

Geometrically, the absolute value of the determinant represents the volumn of a high dimensional parallelogram with ** B**-paralleogram as "base" and

The triangle property can be generalized (by induction and transpose, for example). For square matrices **A**_{1}, **A**_{2}, ..., ` A_{k}`, we have

det[ |
A_{1} |
# |
. . | # |
] = detA_{1} detA_{2} ... det,A_{k} |

O |
A_{2} |
. . | # |
||

: | : | : | |||

O |
O |
. . | A_{k} |

det[ |
A_{1} |
O |
. . | O |
] = detA_{1} detA_{2} ... det.A_{k} |

# |
A_{2} |
. . | O |
||

: | : | : | |||

# |
# |
. . | A_{k} |

In particular, for upper and lower triangular matrices, we have

det[ |
a_{1} |
# | . . | # | ] = a_{1}a_{2}...a,_{n} |

0 | a_{2} |
. . | # | ||

: | : | : | |||

0 | 0 | . . | a_{n} |

det[ |
a_{1} |
0 | . . | 0 | ] = a_{1}a_{2}...a._{n} |

# |
a_{2} |
. . | 0 | ||

: | : | : | |||

# | # | . . | a_{n} |

Further specializing to the identity matrix, we have

`det I` = 1.

Geometrically, this simply means that the cube with unit side length has volume 1, and the standard basis has the positive orientation. Moreover, from `AA`^{-1} = ** I**, we have

`det`(` A^{k}`) = (

Example

det[ |
1 | 0 | 2 | 0 | ] = -det[ |
1 | 2 | 0 | 0 | ] = det[ |
1 | 2 | 0 | 0 | ] | |||||||

0 | 3 | 0 | 4 | 0 | 0 | 3 | 4 | 5 | 6 | 0 | 0 | = det[ |
1 | 2 | ] det[ |
3 | 4 | ] = (-4)(-4) = 16. | ||||

5 | 0 | 6 | 0 | 5 | 6 | 0 | 0 | 0 | 0 | 3 | 4 | 5 | 6 | 7 | 8 | |||||||

0 | 7 | 0 | 8 | 0 | 0 | 7 | 8 | 0 | 0 | 7 | 8 |

Example Let ** A** and

`det`(` AB`) =

Finally, we remark that

`det`(** A** +

However, the determinant is multilinear with respect to rows and columns.

In `d A`, the number

`det`(`d A`) =