0
votes
1answer
24 views

volume of parallelotopes

I know that determinant indicates the volume of a parallelotopes spanned by the n vectors. I absolutely understand that the properties of a determinant: any function $f:\mathbb{R}^{n\times ...
2
votes
2answers
69 views

Geometric interpretation of determinant

I am trying to prove geometrically, without invoking the dot or cross products or orthogonality, that the volume of a parallelepiped formed by vectors $ \begin{bmatrix} a_1 \\ a_2 \\ a_3 ...
4
votes
1answer
145 views

Cramer's rule: Geometric Interpretation

I have a question concerning Cramer's rule: Let $A$ be a matrix and $A \cdot \vec x = \vec b$ a lineare equation. $A_i$ is the matrix $A$ where the i'th column is replaced by $\vec b$ if $det(A) ...
18
votes
3answers
737 views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
1
vote
1answer
35 views

Geometric meaning of minors

This is a bit silly question I found on another discussion forum. I know that determinants can be used to compute volumes of parallelepiped. I also know that determinants can be computed by linear ...
4
votes
3answers
2k views

why determinant is volume of parallelepiped in any dimensions

for $n = 2,$ I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculate the area by coordinates. But how can one easily realize that it is true ...
2
votes
3answers
143 views

On integral of a function over a simplex

Help w/the following general calculation and references would be appreciated. Let $ABC$ be a triangle in the plane. Then for any linear function of two variables $u$. $$ \int_{\triangle}|\nabla ...
3
votes
2answers
238 views

Parallelogram area using determinant

Given a Parallelogram with the co-ordinates: $(a+c, b+d), (c,d), (a, b)$ and $(0, 0)$ I have to prove that the area of the Parallelogram is: $|ad-bc|$ as in the determinant of: $$\begin{bmatrix} a ...
4
votes
1answer
38 views

Determinant of the matrix $D_n(2,3,1)$

The matrix $D_n(2,3,1)$ is to be written in the form $$\pmatrix{3 & 1 & 0 & 0 & ... & 0 \\ 2 & 3 & 1 & 0 & ... & 0 \\ 0 & 2 & 3 & 1 &... ...
1
vote
2answers
242 views

How to workout the determinant of the matrix $D_n(\alpha, \beta, \gamma)$.

I am going through an example in my lecture notes. This is it: Let's introduce the matrix $D_n(\alpha, \beta, \gamma)$, which looks like this: $$\pmatrix{\beta & \gamma & 0 & 0 ...
2
votes
1answer
146 views

Some Questions on Determinants and Geometry

For real valued matrices, I know that the absolute value of the determinant is equivalent to the volume of the vectors forming the parallelepiped in the matrix. Suppose that $A$ and $B$ are real ...
1
vote
1answer
762 views

Given a parallelepiped, how do I find the determinant given vertices?

Here are the given vertices of a given parallelepiped... $ (-1, 0, 0), (0, 4, 0), (-3, -5, 2), (-2, 2, -1) $ I know that first, we should translate all to the origin... $ (0, 0, 0), (1, 4, 0), (-2, ...
1
vote
2answers
280 views

Determinant form of equation, 3 variables, third order (nomogram)

I'm trying to put the following equation in determinant form: $12h^3 - 6ah^2 + ha^2 - V = 0$, where $h, a, V$ are variables (this is a volume for a pyramid frustum with $1:3$ slope, $h$ is the height ...
1
vote
1answer
347 views

Meaning of this 4x4 determinant

Let $p,q,r$ and $s$ be four points on the plane. Moreover, $p,q,r$ are given in clockwise order. My book said that the following determinant is positive if and only if $s$ lies inside the circle ...
0
votes
1answer
285 views

volume of a parallelogram [duplicate]

Possible Duplicate: Determinants and volume of parallelotopes Can you give me a direction about how to prove that $|det(UVW)|$ is the 3D volume of the parallelogram that defined by $U, V$ ...
3
votes
2answers
185 views

How do I prove that the following method to find whether a point lies within a polygon is correct?

I came across the following method to determine whether a given point lies inside a convex polygon - however, I'm not sure how to prove it. Given any three points on the plane $(x_0,y_0)$, ...