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Chapter 1: Problem 1

Berechnen Sie die Determinanten der folgenden 2-reihigen Matrizen: a) \(A=\left(\begin{array}{rr}2 & 3 \\ 4 & -5\end{array}\right)\) b) \(\quad B=\left(\begin{array}{ll}a & a \\ b & b\end{array}\right)\) c) \(C=\left(\begin{array}{ll}3 & 11 \\ x & 2 x\end{array}\right)\)

Short Answer

Expert verified
The determinants are: (a) -22, (b) 0, (c) -5x.

Step by step solution

01

- Determinant Formula for 2x2 Matrices

The determinant of a 2x2 matrix the matrix is given by the formula \[\text{det}(M) = ad - bc\]. where are the entries of the matrix.
02

- Calculate Determinant of Matrix A

For matrix \[A=\begin{pmatrix} 2 & 3 \ 4 & -5 \end{pmatrix}\], applying the determinant formula: \[a = 2,\ b = 3,\ c = 4,\ d = -5\] gives \[\text{det}(A) = 2 \times (-5) - 3 \times 4 = -10 - 12 = -22\].
03

- Calculate Determinant of Matrix B

For matrix \[B=\begin{pmatrix} a & a \ b & b \end{pmatrix}\], applying the determinant formula \[a = a, \ b = a, \ c = b, \ d = b\] gives \[\text{det}(B) = a \times b - a \times b = ab - ab = 0\].
04

- Calculate Determinant of Matrix C

For matrix \[C=\begin{pmatrix} 3 & 11 \ x & 2x \end{pmatrix}\], applying the determinant formula: \[a = 3, \ b = 11, \ c = x, \ d = 2x\] gives \[\text{det}(C) = 3 \times (2x) - 11 \times x = 6x - 11x = -5x\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Determinant
To understand matrix determinants, let's start with a 2x2 matrix. A 2x2 matrix has the form \[M = \begin{pmatrix} a & b \ c & d \ \end{pmatrix} \].
The determinant of a 2x2 matrix is calculated using the formula: \[ \text{det}(M) = ad - bc \].
Here, 'a', 'b', 'c', and 'd' are the elements of the matrix. The determinant gives us vital information about the matrix, such as whether the matrix is invertible or not.
For a matrix to be invertible, its determinant should not be zero. Now, let's calculate the determinants for given matrices using this formula.
Linear Algebra
Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. It's widely used in various fields like physics, computer science, and engineering.
Understanding the determinant is a key part of linear algebra. The determinant helps in many applications:
  • Solving systems of linear equations.
  • Determining matrix invertibility.
  • Finding eigenvalues and eigenvectors.
Linear algebra provides the tools to work with multi-dimensional data, making it essential for understanding how high-dimensional spaces function and interact.
Matrix Calculation
Matrix calculations involve operations such as addition, subtraction, multiplication, and finding the determinant. Let's dive into the calculations for the provided matrices:
  • **Matrix A**: \[ A = \begin{pmatrix} 2 & 3 \ 4 & -5 \ \end{pmatrix} \]. Using the formula: \[ \text{det}(A) = 2 \times (-5) - 3 \times 4 = -10 - 12 = -22 \].
  • **Matrix B**: \[ B = \begin{pmatrix} a & a \ b & b \ \end{pmatrix} \]. Applying the formula: \[ \text{det}(B) = a \times b - a \times b = 0 \].
  • **Matrix C**: \[ C = \begin{pmatrix} 3 & 11 \ x & 2x \ \end{pmatrix} \]. Calculating the determinant: \[ \text{det}(C) = 3 \times (2x) - 11 \times x = 6x - 11x = -5x \].
These calculations show how the determinant formula is applied to different matrices.

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Most popular questions from this chapter

Aus Band 1 , Abschnitt II.3.5 ist bekannt: Das Spatprodukt \([\vec{a} \vec{b} \vec{c}]=\vec{a} \cdot(\vec{b} \times \vec{c})\) dreier Vektoren aus dem Anschauungsraum kann aus den skalaren Vektorkomponenten mit Hilfe der Determinante $$ [\vec{a} \vec{b} \vec{c}]=\left|\begin{array}{lll} a_{x} & a_{y} & a_{z} \\ b_{x} & b_{y} & b_{z} \\ c_{x} & c_{y} & c_{z} \end{array}\right| $$ berechnet werden. Zeigen Sie, daB die drei Vektoren $$ \vec{a}=\left(\begin{array}{l} 1 \\ 2 \\ 2 \end{array}\right), \quad \vec{b}=\left(\begin{array}{r} 0 \\ -4 \\ 3 \end{array}\right) \text { und } \vec{c}=\left(\begin{array}{r} 3 \\ -6 \\ 15 \end{array}\right) $$ komplanar sind, d.h. in einer Ebene liegen.

F?r welchen Wert des Parameters \(\lambda\) sind die Vektoren $$ \mathbf{a}_{1}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right), \quad \mathbf{a}_{2}=\left(\begin{array}{r} -\lambda \\ 2 \\ 5 \end{array}\right) \text { und } \mathbf{a}_{3}=\left(\begin{array}{r} -2 \\ \lambda \\ 7 \end{array}\right) $$ linear abh?ngig?

Matrix A beschreibt die Spiegelung eines Raumpunktes an der \(x, y\)-Ebene, Matrix B die Drehung des r?umlichen Koordinatensystems um die \(z\)-Achse um den Winkel \alpha. Zeigen Sic, daB beide Matrizen orthogonal sind. $$ \mathbf{A}=\left(\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array}\right), \quad B=\left(\begin{array}{ccc} \cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{array}\right) $$

Welche der folgenden 3-rcihigen Matrizen sind orthogonal? $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & 0 & 1 \\ 2 & -2 & 0 \\ 0 & -1 & 3 \end{array}\right), \quad \mathrm{B}=\frac{1}{3}\left(\begin{array}{rrr} 2 & 2 & 1 \\ 1 & -2 & 2 \\ 2 & -1 & -2 \end{array}\right) \\ &C=\left(\begin{array}{ccc} 0 & 1 / \sqrt{2} & -1 / \sqrt{2} \\ 0 & 1 / \sqrt{2} & 1 / \sqrt{2} \\ 1 & 0 & 0 \end{array}\right) \end{aligned} $$

Lösen Sie die folgenden nicht-quadratischen linearen Gleichungssysteme mit Hilfe elementarer Umformungen in den Zeilen der erweiterten Koeffizientenmatrix (Gau\betascher Algorithmus): \(2 x_{1}-3 x_{2}=11\) a) \(-5 x_{1}+x_{2}=-8\) b) \(\left(\begin{array}{rrrr}1 & 1 & 2 & 0 \\ -3 & 2 & 0 & 1 \\ 8 & -2 & -2 & 2\end{array}\right)\left(\begin{array}{c}x_{1} \\ x_{2} \\ x_{3} \\\ x_{4}\end{array}\right)=\left(\begin{array}{l}1 \\ 5 \\ 0\end{array}\right)\) \(x_{1}-5 x_{2}=16\) \(3 x_{2}-5 x_{3}+x_{4}=0\) c) $$ \begin{aligned} &-x_{1}-3 x_{2}-x_{4}=-5 \\ &-2 x_{1}+x_{2}+2 x_{3}+2 x_{4}=2 \\ &-3 x_{1}+4 x_{2}+2 x_{3}+2 x_{4}=8 \end{aligned} $$
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