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Chapter 4: Problem 9

Bestimmen Sie den Rang der folgenden reellen \(n \times n\)-Matrix: $$ \left(\begin{array}{ccccc} a & b & b & \ldots & b \\ b & a & b & \ldots & b \\ b & b & a & \ldots & b \\ \cdots & & \cdots & & \cdots \\ b & b & b & \ldots & a \end{array}\right) $$ Dabei sind \(a\) und \(b\) verschiedene reelle Zahlen. [Hinweis: Subtrahieren Sie zun盲chst die erste Zeile von allen anderen Zeilen.]

Short Answer

Expert verified
The rank of the matrix is \( n \).

Step by step solution

01

Understand the Matrix Structure

The given matrix is an \( n \times n \) matrix where \( a \) appears as the diagonal elements and \( b \) appears in all other positions. The matrix is symmetric and is of the following form: \[ \begin{pmatrix} a & b & b & \cdots & b \ b & a & b & \cdots & b \ b & b & a & \cdots & b \ \vdots & & \vdots & & \vdots \ b & b & b & \cdots & a \end{pmatrix} \] with \( a eq b \).
02

Perform Row Operations

According to the hint provided, subtract the first row from all subsequent rows. This means performing the operation \( R_i = R_i - R_1 \) for each row \( i \) (where \( i = 2, 3, \ldots, n \)).
03

Evaluate the Resulting Matrix

After executing the row operations, the matrix transforms such that the first row remains unchanged: \( (a, b, b, \ldots, b) \). The remaining \( (n-1) \) rows become \( (0, a-b, 0, \ldots, 0) \) for the second row, \( (0, 0, a-b, \ldots, 0) \) for the third row, and similarly for the others. This results in a diagonal matrix beneath the first row, where each diagonal entry is \( a-b \).
04

Determine the Rank of the Matrix

The first row is non-zero, contributing to the rank. The diagonal entries below are all \( a-b \), which is non-zero since \( a eq b \). This means there are exactly \( (n-1) \) linearly independent non-zero rows formed by these operations. Therefore, the rank of the matrix is \( n \).

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Key Concepts

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

Symmetric Matrix
A symmetric matrix is a square matrix that remains unchanged when transposed. In other words, the matrix looks the same whether you read it left to right across the rows or up and down the columns. In mathematical terms, a matrix \( A \) is symmetric if \( A = A^T \), where \( A^T \) represents the transpose of \( A \). This means that each element in the matrix is mirrored across its main diagonal. For instance, in a 3x3 symmetric matrix, the element at the first row, second column is the same as the element at the second row, first column. Interesting Properties:
  • All elements on the main diagonal are unrestricted and can be any number.
  • Symmetric matrices are often straightforward to analyze due to their mirrored properties.
  • They frequently appear in mathematical problems involving distance or similarity calculations, where symmetry is typical.
Row Operations
Row operations are fundamental in matrix algebra and are powerful tools for simplifying matrices to make them easier to work with, especially in solving systems of linear equations or determining matrix rank. There are three main types of row operations:
  • Swapping two rows: Changes the order but not the content.
  • Scalar multiplication: Multiply all elements in a row by a non-zero number.
  • Row addition: Add or subtract a multiple of one row to another. This transforms the matrix but doesn't alter the overall solution to the system.
These operations are useful in transforming a matrix into a simpler form, like a diagonal or reduced row-echelon form. For instance, in our exercise, subtracting the first row from all other rows helped isolate some of the matrix's features and made it easier to discern its rank.
Linear Independence
Linear independence is a key idea in linear algebra that describes a set of vectors that do not overlap in their span. In simpler terms, a set of vectors is linearly independent if no vector can be written as a combination of the others. If at least one vector can be expressed as a combination of others, the vectors are linearly dependent.How It Works:
  • An equation \( c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \ldots + c_n\mathbf{v}_n = \mathbf{0} \) reaches a trivial solution where all \( c_i = 0 \) only if the vectors are independent.
  • In the context of matrices, row operations can help identify independent rows, which directly affect the rank of the matrix.
In the matrix from the exercise, the transformation through row operations resulted in multiple independent diagonal entries, confirming the non-dependence of rows and ultimately determining the matrix's rank.
Diagonal Matrix
A diagonal matrix is a type of square matrix in which all the non-diagonal elements are zero. Only the elements along the main diagonal can be non-zero. This is expressed as \( D = \text{diag}(d_1, d_2, \ldots, d_n) \), meaning: \[ \begin{pmatrix} d_1 & 0 & \cdots & 0 \0 & d_2 & \cdots & 0 \\vdots & \vdots & \ddots & \vdots \0 & 0 & \cdots & d_n \end{pmatrix} \]Why Are They Useful?
  • Computation involving diagonal matrices is often easier as you only need to consider the main diagonal elements.
  • The determinant of a diagonal matrix is simply the product of its diagonal elements.
  • Diagonal matrices appear in the study of eigenvalues and eigenvectors, often simplifying these topics.
In this exercise, after performing certain row operations, a diagonal matrix emerged in the matrix's lower section, helping to easily assess linearly independent rows and calculate the rank.

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Most popular questions from this chapter

Sei \(V\) ein \(K\)-Vektorraum. Wir nennen eine Linearkombination \(k_{1} v_{1}+k_{2} v_{2}+\ldots+\) \(k_{n} v_{n}\) eine affine Linearkombination, falls \(k_{1}+k_{2}+\ldots+k_{n}=1\) ist. Zeigen Sie: Jede affine Linearkombination von Vektoren eines affinen Unterraums \(U\) liegt ebenfalls in \(U\)

Zeigen Sie, dass sich der L枚sungsraum eines linearen Gleichungssystems bei elementaren Zeilenumformungen vom Typ 3 nicht 盲ndert.

Welchen Rang haben die folgenden reellen Matrizen? $$ \left(\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right),\left(\begin{array}{cccc} 1 & 2 & 3 & 0 \\ 4 & 5 & 6 & 0 \\ 7 & 8 & 9 & 0 \\ 10 & 11 & 12 & 0 \end{array}\right),\left(\begin{array}{cccc} 1 & 2 & 3 & 0 \\ 7 & 2 & 0 & 3 \\ 1 & -1 & 3 & -8 \\ -1 & 5 & 3 & 8 \end{array}\right) $$

Ein Code \(C \subseteq\\{0,1\\}^{n}\) hei脽t \(s\)-fehlererkennend, falls kein Codewort aus \(C\) durch Hinzuf眉gen von h枚chstens \(s\) Fehlern wieder zu einem Codewort aus \(\mathrm{C}\) wird. (a) Machen Sie sich klar, dass dieser Name zu Recht besteht. (b) Zeigen Sie: Ein Code \(\mathbf{C}\) ist genau dann \(s\)-fehlererkennend, wenn \(d(\mathbf{C}) \geq s+1\) ist.

Bestimmen Sie den Rang der folgenden reellen Matrizen \(A, B\) und \(A \cdot B\) : $$ A=\left(\begin{array}{cccc} 3 & 5 & 1 & 4 \\ 2 & -1 & 1 & 1 \\ 8 & 9 & 3 & 9 \end{array}\right), \quad B=\left(\begin{array}{cccc} 2 & 1 & 6 & 6 \\ 3 & 1 & 1 & -1 \\ 5 & 2 & 7 & 5 \\ -2 & 4 & 3 & 2 \end{array}\right) $$
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