### Geometry of Determinant

##### 3. Determinant and volumn change

The geometrical interpretation of determinant leads to another interpretation as the *ratio for the change of volume* under the transformation corresponding to the matrix. A detailed explaination for the fact can be found here.

Let `A` be an `n` by `n` matrix. Let `R` be a region in **R**^{n}, and **A**R be the image under the transformation given by `A`. Then

(volume of **A**R) = |`det`**A**| (volume of `R`).

The relation between the determinant and the change of volume is the reason behind the use of Jacobian (which is the absolute value of the determinant of the first order derivative matrix) in the change of variable formula in multivariable integration.

Example In an earlier example, we computed the area of the triangle `R` with vertices (1, 2), (7, 1), (3, 5). The conclusion was area(`R`) = 10. If we apply the linear transformation by the matrix

then the image `A``R` is a triangle with vertices (0, 3), (13, 8), (1, 8).

As in the earlier example, we have

area(`A``R`) = (1/2)| `det`[ |
13 - 0 |
1 - 0 |
] | = 30. |

8 - 3 |
8 - 3 |

On the other hand, we have `det`**A** = 3, and the equality area(`A``R`) = |`det`**A**| area(`R`) indeed holds.

Example We would like to compute the area of the ellipse 9`x`^{2} + 6`xy` + 5`y`^{2} ≤ 36. The inequality is the same as (3`x` + `y`)^{2} + (2`y`)^{2} ≤ 6^{2}. Thus the ellipse becomes the unit disk `w`^{2} + `z`^{2} ≤ 1 under the transformation (`x`, `y`) ↔ (`w`, `z`) given by `w` = (3`x` + `y`)/6, `z` = 2`y`/6.

Alternatively, we may consider the ellipse as the image of the unit disk under the transformation (`w`, `z`) → (`x`, `y`) = (2`w` - `z`, 3`z`). Thus

area(ellipse) = | `det`[ |
2 |
-1 |
] | area(unit disk) = 6π. |

0 |
3 |

The sign of the determinant also has interpretation in terms of linear transformations. For a 2 by 2 matrix `A`, we say

`A` preserves orientation if `u` to `v` is clockwise (counterclockwise) ⇒ `Au` to `Av` is clockwise (counterclockwise);

`A` reverses orientation if `u` to `v` is clockwise (counterclockwise) ⇒ `Au` to `Av` is counterclockwise (clockwise).

Then we have (see here for explanation)

`A` preserves orientation ⇔ `det`**A** ≥ 0;

`A` reverses orientation ⇔ `det`**A** ≤ 0.

The similar discussion applies to 3 dimensional case. A 3 by 3 matrix preserves orientation if it takes left (right) hand arrangement to left (right) hand arrangement. In this case the determinant is ≥ 0. The matrix reverses orientation if it takes left (right) hand arrangement to right (left) hand arrangement. In this case the determinant is ≤ 0.

##### [exercise]

[part 1]
[part 2]
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[Extra The determinant and self linear transformation of R^{2}]