Problem 32 - A History of Mathematics: An Introduction

Question

Find a nilpotent element in the algebra of \(2 \times 2\) matrices over the rational numbers

Step-by-Step Solution

Answer

Question: Find a nilpotent element in the 2x2 matrix algebra over the rational numbers. Answer: A nilpotent element in the 2x2 matrix algebra over the rational numbers is the matrix A = $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$.

Step 1: Definition of nilpotent element

A nilpotent element (matrix) is an element \(A \in M_2(\mathbb{Q})\), where \(M_2(\mathbb{Q})\) is the algebra of \(2 \times 2\) matrices over the rational numbers, such that \(A^n = 0\) for some positive integer \(n\). Our task is to find such a matrix.

Step 2: Trial and error approach

Since there are infinitely many matrices, trial and error can be a potential approach to find a nilpotent matrix. Let's try a simple matrix: $$ A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} $$ Now, let's find the square of the matrix A.

Step 3: Calculate the square of A

Calculate the square of matrix A: $$ A^2 = A \cdot A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} $$

Step 4: Matrix multiplication

Use the matrix multiplication: $$ A^2 = \begin{pmatrix} 0\cdot0 + 1\cdot0 & 0\cdot1 + 1\cdot0 \\ 0\cdot0 + 0\cdot0 & 0\cdot1 + 0\cdot0 \end{pmatrix} $$

Step 5: Simplify the result

Simplify the resulting matrix: $$ A^2 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} $$ We found that \(A^2\) is the zero matrix.

Step 6: Conclusion

Since for the matrix A = $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\(, we have \)A^2 = 0$, with 0 being the zero matrix. Therefore, matrix A is a nilpotent element in the algebra of 2x2 matrices over the rational numbers.

Key Concepts

Linear Algebra and Nilpotent Matrices

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and operations that can be applied to these vectors and spaces, like addition and scalar multiplication. Among these operations, one particularly interesting case is that of matrices, which can be thought of as a tabular array of numbers.

One intriguing type of matrix encountered in linear algebra is the nilpotent matrix. A nilpotent matrix is defined as a square matrix that, when multiplied by itself a certain number of times, results in the zero matrix, symbolically denoted as \( A^n = 0 \). In the given exercise, we were tasked with finding a \( 2 \times 2 \) nilpotent matrix over the rational numbers, which are fractions or numbers that can be expressed as the quotient of two integers. The solution demonstrated effectively how to identify such a matrix through an elementary example, reinforcing the concept within a linear algebra framework.

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra where two matrices are multiplied to obtain another matrix. The process involves taking the rows of the first matrix and columns of the second matrix and computing the sum of the products of their respective entries.

To visualize this, imagine two \( 2 \times 2 \) matrices being multiplied together. Each entry in the resulting matrix is computed by adding the product of the corresponding elements from the row of the first matrix and the column of the second matrix—this was executed step-by-step in the exercise. The example given demonstrates that even with rational numbers, the rules for matrix multiplication apply the same way. Matrix multiplication is not commutative, meaning that in general, \( A \cdot B \) does not equal \( B \cdot A \). However, if \( A \) is nilpotent, like the matrix in the exercise, then any power of \( A \) beyond the nilpotent index results in a zero matrix, regardless of the multiplication order.

Rational Numbers in Matrices

Rational numbers are numbers that can be expressed as a ratio between two integers where the denominator is not zero. In linear algebra, matrices can be filled with any kind of number, including rational numbers. These numbers are essential because they allow us to perform accurate and precise calculations in various mathematical and real-world scenarios.

In the context of our exercise, the rational numbers were used to fill the entries of a \( 2 \times 2 \) matrix. It was essential to understand that the concept of nilpotent matrices applies over any field, including the field of rational numbers. When we square the matrix \( A \) containing rational numbers, if it results in a zero matrix, then \( A \) is nilpotent. This property of matrices does not change with the type of number used in their entries, which underscores the versatility and broad applicability of linear algebra in various domains, from pure mathematics to engineering and social sciences.