Question

Man invertiere die Matrix \([\mathrm{M}]:=\left[\begin{array}{lll}1 & \mathrm{x}_{1} & \mathrm{y}_{1} \\ 1 & \mathrm{x}_{2} & \mathrm{y}_{2} \\ 1 & \mathrm{x}_{3} & \mathrm{y}_{3}\end{array}\right]\) (4.13) und zeige, \(\mathrm{daB}\) ihre Determinante mit dem doppelten Flächeninhalt des finiten Dreiecks übereinstimmt, das die Eckpunkte \(\left(x_{1}, y_{1}\right), \ldots \ldots,\left(x_{3}, y_{3}\right)\) besitzt.

Short answer

The determinant of \(\textbf{M}\) equals twice the area of the triangle.

Step-by-step solution

Write the Given Matrix

The matrix \(\textbf{M}\) is given as: \[ \textbf{M} = \begin{bmatrix} 1 & x_1 & y_1 \ 1 & x_2 & y_2 \ 1 & x_3 & y_3 \end{bmatrix} \]

Compute the Determinant

To find the determinant of the matrix \(\textbf{M}\), use the formula for the determinant of a 3x3 matrix. The determinant \( \text{det}(\textbf{M}) \) is given by: \[ \text{det}(\textbf{M}) = 1 \begin{vmatrix} x_2 & y_2 \ x_3 & y_3 \end{vmatrix} - x_1 \begin{vmatrix} 1 & y_2 \ 1 & y_3 \end{vmatrix} + y_1 \begin{vmatrix} 1 & x_2 \ 1 & x_3 \end{vmatrix} \]

Compute the 2x2 Determinants

Calculate the determinants of the 2x2 matrices: \[ \begin{vmatrix} x_2 & y_2 \ x_3 & y_3 \end{vmatrix} = x_2 y_3 - x_3 y_2 \] \[ \begin{vmatrix} 1 & y_2 \ 1 & y_3 \end{vmatrix} = y_2 - y_3 \] \[ \begin{vmatrix} 1 & x_2 \ 1 & x_3 \end{vmatrix} = x_2 - x_3 \]

Substitute Back Into Determinant Formula

Substitute the computed 2x2 determinants: \[ \text{det}(\textbf{M}) = 1(x_2 y_3 - x_3 y_2) - x_1(y_2 - y_3) + y_1(x_2 - x_3) \]

Simplify the Expression

Combine and simplify the terms: \[ \text{det}(\textbf{M}) = x_2 y_3 - x_3 y_2 - x_1 y_2 + x_1 y_3 + y_1 x_2 - y_1 x_3 \]

Connect to Triangle Area

The area of a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is given by: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \] Notice that this expression matches the simplified determinant of \(\textbf{M}\), up to a factor of 2. Thus, \[ 2 \times \text{Area} = \left| \text{det}(\textbf{M}) \right| \]

Key concepts

Determinant Calculation

The determinant of a matrix is a special number that can be calculated from its elements. For a 3x3 matrix, the determinant is computed using a specific formula that involves the elements of the matrix.

In our problem, we have a 3x3 matrix \textbf{M} given as: \[ \textbf{M} = \begin{bmatrix} 1 & x_1 & y_1 \ 1 & x_2 & y_2 \ 1 & x_3 & y_3 \end{bmatrix} \]

To find its determinant, \text{det}(\textbf{M}), we use the expansion along the first row: \[ \text{det}(\textbf{M}) = 1 \begin{vmatrix} x_2 & y_2 \ x_3 & y_3 \end{vmatrix} - x_1 \begin{vmatrix} 1 & y_2 \ 1 & y_3 \end{vmatrix} + y_1 \begin{vmatrix} 1 & x_2 \ 1 & x_3 \end{vmatrix} \]

Here, the determinants of the 2x2 submatrices are calculated and then substituted back into the formula.

This method of expanding the determinant is very useful for understanding how changes in matrix elements affect the overall determinant.

Finite Elements Method

The Finite Elements Method (FEM) is a computational technique used to find approximate solutions to complex problems in engineering and mathematical physics. It works by breaking down a large system into smaller, simpler parts called finite elements.

Each element, usually a triangle or quadrilateral in 2D problems, is analyzed individually and then combined to represent the whole system.

In the context of our problem, the matrix \textbf{M} is related to the finite element method because it helps in defining the area and various properties of a triangular element. The vertices of this triangle are given by the points \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), and the determinant of the matrix \textbf{M} is used to find important characteristics such as the triangle's area.

Triangle Area

The area of a triangle is a fundamental geometric property that can be calculated using various formulas. For a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), the area can be found using the determinant of a matrix.

The formula for the area of such a triangle is: \[ \text{Area} = \frac{1}{2} \bigg| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \bigg| \]

This formula is very similar to the simplified determinant of the matrix \textbf{M}, as shown in the step-by-step solution:

\[ \text{det}(\textbf{M}) = x_2 y_3 - x_3 y_2 - x_1 y_2 + x_1 y_3 + y_1 x_2 - y_1 x_3 \]

We can see that the area of the triangle is exactly half of the absolute value of this determinant. Hence, the relationship can be written as: \[ 2 \times \text{Area} = \big| \text{det}(\textbf{M}) \big| \]
This shows a clear link between the geometric concept of the triangle's area and the algebraic concept of the matrix determinant.