To solve Aufgabe 13, I will calculate the determinant for each matrix provided:

Aufgabe 13 - Determinante

The task requires finding the determinant of each of the matrices:

1. Matrix A $A = \begin{pmatrix} 4 & 1 \\ -4 & 2 \end{pmatrix}$ Determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by $\det(A) = ad - bc$.

2. Matrix B $B = \begin{pmatrix} 1 & -4 & -10 \\ 10 & -8 & 2 \\ 0 & -1 & 6 \end{pmatrix}$ For $3 \times 3$ matrices, the determinant can be calculated using cofactor expansion.

3. Matrix C $C = \begin{pmatrix} -7 & 1 & -10 \\ 1 & 10 & 2 \\ 1 & 0 & 6 \end{pmatrix}$

4. Matrix D $D = \begin{pmatrix} -3 & 0 & -8 & 7 \\ -7 & 1 & -4 & -10 \\ 1 & 10 & 8 & 2 \\ 1 & 0 & 1 & 6 \end{pmatrix}$ This is a $4 \times 4$ matrix, so the determinant will be computed using cofactor expansion or other methods suitable for larger matrices.

5. Matrix E $E = D^{-1}$ Since matrix $E$ is the inverse of $D$, the determinant of $E$ is the reciprocal of the determinant of $D$: $\det(E) = \frac{1}{\det(D)}$, provided $\det(D) \neq 0$.

6. Matrix F $F = \begin{pmatrix} 3 & 0 & 8 & -7 \\ 7 & -1 & 4 & 10 \\ -1 & -10 & 8 & -2 \\ -1 & 0 & 1 & -6 \end{pmatrix}$ Similar to $D$, the determinant of this $4 \times 4$ matrix can be found using cofactor expansion.

I will now calculate each determinant.import numpy as np

Defining each matrix based on the problem

Matrix A (2x2)

A = np.array([[4, 1], [-4, 2]])

Matrix B (3x3)

B = np.array([[1, -4, -10], [10, -8, 2], [0, -1, 6]])

Matrix C (3x3)

C = np.array([[-7, 1, -10], [1, 10, 2], [1, 0, 6]])

Matrix D (4x4)

D = np.array([[-3, 0, -8, 7], [-7, 1, -4, -10], [1, 10, 8, 2], [1, 0, 1, 6]])

Matrix F (4x4)

F = np.array([[3, 0, 8, -7], [7, -1, 4, 10], [-1, -10, 8, -2], [-1, 0, 1, -6]])

Calculating determinants

det_A = np.linalg.det(A) det_B = np.linalg.det(B) det_C = np.linalg.det(C) det_D = np.linalg.det(D) det_E = 1 / det_D if det_D != 0 else None # Inverse determinant if D is non-singular det_F = np.linalg.det(F)

det_A, det_B, det_C, det_D, det_E, det_F(12.0, 293.99999999999994, -323.9999999999999, 2194.9999999999986, 0.00045558086560364494, 989.0000000000003)(12.0, 293.99999999999994, -323.9999999999999, 2194.9999999999986, 0.00045558086560364494, 989.0000000000003)successHere are the determinants for each matrix:

1. Determinant of A: $\det(A) = 12$
2. Determinant of B: $\det(B) = 294$
3. Determinant of C: $\det(C) = -324$
4. Determinant of D: $\det(D) = 2195$
5. Determinant of E (inverse of D): $\det(E) = 0.000456$ (approximately, as $\det(E) = \frac{1}{\det(D)}$)
6. Determinant of F: $\det(F) = 989$

Would you like further details on any specific calculation, or do you have any questions?

Further Questions:

1. How are determinants used to find the inverse of a matrix?
2. What does a zero determinant signify about a matrix?
3. How can cofactor expansion be applied to calculate determinants of larger matrices?
4. What are some properties of determinants in matrix multiplication?
5. How does determinant scaling work when a matrix is multiplied by a scalar?

Tip:

For a matrix to be invertible, its determinant must be non-zero.