Welchen Wert besitzen die 3-reihigen Determinanten? a) \(\left|\begin{array}{lll}1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9\end{array}\right|\) b) \(\left|\begin{array}{rrr}-2 & 8 & 2 \\ 1 & 0 & 7 \\ 4 & 3 & 1\end{array}\right|\) c) \(\left|\begin{array}{rrr}3 & 4 & -10 \\ -7 & 4 & 1 \\ 0 & 2 & 8\end{array}\right|\) (Berechnung nach der Regel von Sarrus).

When dealing with matrices in linear algebra, one important concept is the determinant. The determinant is a special number that can be calculated from a square matrix. It provides useful information about the matrix. For instance, if the determinant of a matrix is zero, the matrix is said to be singular, meaning it doesn't have an inverse. On the other hand, a non-zero determinant indicates that the matrix is invertible. Determinants can be computed using various methods, offering insight into the matrix's properties. A common method for 3x3 matrices is Sarrus' Rule.

A 3x3 matrix is a square matrix with three rows and three columns. It looks like this: \[A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \](https://latex.codecogs.com/svg.image?A%20%3D%20%5Cbegin%7Bpmatrix%7D%20a%20%26%20b%20%26%20c%20%5C%5C%20d%20%26%20e%20%26%20f%20%5C%5C%20g%20%26%20h%20%26%20i%20%5Cend%7Bpmatrix%7D) To compute the determinant of a 3x3 matrix, you can use Sarrus' Rule. This rule involves a straightforward method where you multiply specific elements and then sum or subtract the results. In general, to find the determinant of matrix A using Sarrus' Rule, you follow these steps:

• Take the product of the diagonal elements from the top-left to bottom-right (i.e. aei).
• Add the products of two more diagonals you can form by shifting one element to the right and wrapping around horizontally if necessary (i.e. bfg and cdh).
• Then subtract the products of the diagonal elements from top-right to bottom-left (i.e. ceg, bdi, afh).

Linear algebra is a branch of mathematics that deals with vectors, matrices, and systems of linear equations. It is pivotal in various scientific and engineering fields. Some key concepts in linear algebra include vector spaces, linear transformations, and eigenvalues. Determinants play a crucial role in linear algebra as they help in understanding the properties of matrices and in solving systems of linear equations. By calculating the determinant, we can determine if a matrix is invertible, solve linear systems, and study linear transformations. Understanding these concepts can significantly aid in numerous applications, such as computer graphics, machine learning, and optimization.