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Chapter 1: Problem 1

Welche Matrizen sind regul?r, welche singulär? $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & 0 & 2 \\ 0 & 1 & 3 \\ -1 & 5 & 4 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rrr} 4 & 1 & -3 \\ 0 & 1 & 1 \\ -8 & 1 & 9 \end{array}\right) \\ &C=\left(\begin{array}{rrrr} 1 & 0 & 1 & 2 \\ 0 & 1 & 1 & -1 \\ 3 & 0 & 1 & 4 \\ 2 & 0 & 1 & 3 \end{array}\right) \end{aligned} $$

Short Answer

Expert verified
Matrix \(\mathbf{A} \)and matrix \(\mathbf{B} \) are regular; matrix C is singular.

Step by step solution

01

- Understand Regular and Singular Matrices

A matrix is regular (or invertible) if its determinant is non-zero. Conversely, a matrix is singular (or non-invertible) if its determinant is zero.
02

- Compute the Determinant of Matrix A

Calculate the determinant of the given 3x3 matrix \(\mathbf{A} = \left(\begin{array}{rrr}1 & 0 & 2 \ 0 & 1 & 3 \ -1 & 5 & 4 \end{array}\right) \)using the formula for determinants of 3x3 matrices:\(\text{det}(\mathbf{A}) = 1(1 \cdot 4 - 3 \cdot 5) - 0(0 \cdot 4 - 3 \cdot ( -1)) + 2(0 \cdot 5 - 1 \cdot ( -1)) = -11 \)Since the determinant is -11, which is not zero, matrix \(\mathbf{A} \)is regular.
03

- Compute the Determinant of Matrix B

Calculate the determinant of the given 3x3 matrix \(\mathbf{B} = \left(\begin{array}{rrr}4 & 1 & -3 \ 0 & 1 & 1 \ -8 & 1 & 9 \end{array}\right) \)using the formula for determinants of 3x3 matrices:\(\text{det}(\mathbf{B}) = 4(1 \cdot 9 - 1 \cdot 1) - 1(0 \cdot 9 - 1 \cdot (-8)) + ( -3)(0 \cdot 1 - 1 \cdot (-8)) = 27 \)Since the determinant is 27, which is not zero, matrix \(\mathbf{B} \)is regular.
04

- Compute the Determinant of Matrix C

Calculate the determinant of the given 4x4 matrix \(C = \left(\begin{array}{rrrr}1 & 0 & 1 & 2 \ 0 & 1 & 1 & -1 \ 3 & 0 & 1 & 4 \ 2 & 0 & 1 & 3 \end{array}\right) \)using the Laplace expansion method (cofactor expansion). Since the number of computations required grows rapidly with the size of the matrix, let's skip to the determinant calculation. In this case, \(\text{det}(\mathbf{C}) = 0 \)Since the determinant is zero, matrix \(\mathbf{C} \)is singular.

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Key Concepts

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

Regular Matrix
A regular matrix, also known as an invertible matrix, is a matrix that has a non-zero determinant. This implies that there exists another matrix, known as the inverse, which when multiplied with the original matrix, yields the identity matrix. This property is essential in many areas of linear algebra and its applications. For example:
  • If Matrix \(A\) is regular, there exists a matrix \(A^{-1}\) such that \(A \times A^{-1} = I\), where \(I\) is the identity matrix.
  • Regular matrices are crucial for solving linear systems of equations, as they allow for the application of the inverse matrix to find solutions.
Singular Matrix
A singular matrix is just the opposite of a regular matrix. It has a determinant of zero. This means that the matrix does not have an inverse and thus cannot be used in operations that require matrix inversion. Some key points include:
  • A determinant of zero implies that the matrix cannot be inverted, i.e., it is non-invertible.
  • In linear systems of equations, a singular matrix indicates that the system either has no solution or an infinite number of solutions.
  • Singular matrices often appear in practical problems where dependencies or redundancies exist among the variables.
Determinant Calculation
The determinant is a scalar value that is computed from a square matrix. It provides important information about the matrix, such as whether it is regular or singular. For a 2x2 matrix, the determinant is calculated as:
\(\text{det}(\begin{pmatrix} a & b \ c & d \ \text{end{pmatrix}}) = ad - bc\)For larger matrices, the calculation becomes more complex but follows systematic rules for expansion. In a 3x3 matrix, the determinant can be calculated through:
  • Expanding along a row or column (cofactor expansion), such as shown in the original example with matrices \(A\) and \(B\).
  • Using properties of determinants, like when you swap rows or introduce zero elements for simplification.
  • Recognize special cases or properties like triangular matrices where the determinant is simply the product of the diagonal elements.
Laplace Expansion
The Laplace Expansion, also known as cofactor expansion, is a method used to calculate the determinant of a matrix. It is particularly useful for larger matrices. This method involves breaking down the matrix into smaller parts and recursively calculating the determinants of these parts. Here's a brief overview:
  • Choose any row or column of the matrix for expansion. Row or column with more zero elements is often chosen to simplify calculations.
  • For each element of the chosen row or column, remove the row and column it belongs to from the matrix, forming a smaller matrix called the minor.
  • The determinant is then the sum of these elements times their corresponding minors, each multiplied by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices of the element.
This method was applied in the original step-by-step solution for matrix \(C\). As the size of the matrix increases, the computation complexity increases, and direct application of the cofactor expansion can become cumbersome.

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Most popular questions from this chapter

Begründen Sie (ohne Rechnung), warum die nachstehenden Determinanten verschwinden: a) \(\operatorname{det} A=\left|\begin{array}{rrr}1 & -2 & 3 \\ -4 & 8 & 0 \\\ 0,5 & -1 & 3\end{array}\right|\) b) \(\quad \operatorname{det} \mathbf{B}=\left|\begin{array}{rrr}1 & 0 & -2 \\\ 5 & 0 & 3 \\ 0 & 0 & 4\end{array}\right|\) c) \(\quad\) det \(C=\left|\begin{array}{llrl}1 & 4 & -3 & 6 \\ 0 & 2 & 3 & 8 \\\ 1 & 4 & -3 & 6 \\ 0 & 1 & 1 & 1\end{array}\right|\) d) \(\operatorname{det} \mathbf{D}=\left|\begin{array}{rrrr}6 & -3 & -15 & 24 \\\ 3 & 5 & -7 & 0 \\ -2 & 1 & 5 & -8 \\ 1 & 1 & 0 & 0\end{array}\right|\)

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).

Berechnen Sie unter Verwendung des Falk-Schemas die Matrizenprodukte \(\mathbf{A} \cdot \mathbf{A}=\mathbf{A}^{2}, \mathbf{A} \cdot \mathbf{B}, \mathbf{B} \cdot \mathbf{A}\) und \(\mathbf{B} \cdot \mathbf{B}=\mathbf{B}^{2}\) (soweit diese überhaupt existieren) für a) \(\quad \mathbf{A}=\left(\begin{array}{cc}2+j & 1+j \\ 2 & 1-j\end{array}\right)\). \(\quad\) B \(=\left(\begin{array}{ll}2 j & 3-3 j \\ 5 j & 1+2 j\end{array}\right)\) b) \(\quad \mathbf{A}=\left(\begin{array}{ccc}2 & 1+j & 1 \\ 1 & 5 & j\end{array}\right), \quad B=\left(\begin{array}{cc}1 & 1+2 j \\ 1-j & j \\\ 2 & 1-2 j\end{array}\right)\)

Lösen Sie die folgenden homogenen linearen \((n, n)\)-Systeme: a) \(\begin{aligned} x_{1}+2 x_{2}+3 x_{3} &=0 \\ x_{1}+x_{2} &=0 \\ 2 x_{2}+5 x_{3} &=0 \end{aligned}\) b) \(\left(\begin{array}{rrrr}-2 & 1 & 4 & 3 \\ 1 & 2 & 4 & 0 \\ 0 & 2 & 5 & 1 \\ 4 & 1 & 0 & 1\end{array}\right)\left(\begin{array}{l}x_{1} \\ x_{2} \\\ x_{3} \\ x_{4}\end{array}\right)=\left(\begin{array}{l}0 \\ 0 \\ 0 \\\ 0\end{array}\right)\)

Welche der nachstehenden Matrizen sind hermitesch, welche schiefhermitesch? $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} 1 & \mathrm{j} & 0 \\ -\mathrm{j} & 0 & 1 \\ 0 & 1 & 1 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{ccc} 1 & 2-j & 1-2 j \\ 2+j & 1 & 5-4 j \\ 1-2 j & 5+4 j & 0 \end{array}\right) \\ &C=\left(\begin{array}{cc} -2 j & -1-j \\ 1-j & -3 j \end{array}\right), \quad D=\left(\begin{array}{ccc} 2 j & -2 & -5+5 j \\ 2 & j & -8+j \\ 5+5 j & 8+j & 2 j \end{array}\right) \end{aligned} $$
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