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Chapter 1: Problem 126

Suppose you pick the one million entries of a \(1000 \times 1000\) matrix independently and at random from the set of digits. Is the determinant of the resulting matrix more likely to be even or odd?

Short Answer

Expert verified
The determinant of the \(1000 \times 1000\) matrix is more likely to be even.

Step by step solution

01

- Understand the problem

We need to determine if the determinant of a randomly filled\(1000 \times 1000\) matrix with digits (0-9) is more likely to be even or odd.
02

- Consider matrix entries

Each entry in the matrix is filled with a digit from 0 to 9. Digits 0, 2, 4, 6, and 8 are even, while 1, 3, 5, 7, and 9 are odd. Thus, there's a 50% chance for any entry to be even, and a 50% chance for it to be odd.
03

- Probability of being even

Recall that a matrix is even if its determinant is an even number. For a determinant to be even, at least one row or one column must have an even sum. Since each digit in any row/column has a 50% chance to be even, it is highly probable that at least one row or column will have an even sum.
04

- Probability of being odd

For the determinant to be odd, every row and every column must have an odd sum. This is highly unlikely in a large matrix like the \(1000 \times 1000\) one because the individual entries are chosen independently and at random, which makes it very likely for at least one row or column to end up with an even sum.
05

- Conclusion

Given the above reasoning, the matrix determinant is significantly more likely to be even. The larger the matrix, the higher the probability of having at least one row or column with an even sum, leading to an even determinant.

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Key Concepts

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

Matrix Probability
When we talk about matrix probability, we often think about the likelihood of certain properties in matrices generated with random entries. In the case of a randomly filled \(1000 \times 1000\) matrix, each entry is chosen independently from the digits 0 to 9. This means that every entry in the matrix will have an equal probability of being even or odd. This randomness plays a crucial role in determining the properties of the matrix, like its determinant.
Even and Odd Determinants
The concept of even and odd determinants relates to the sum of digits within the matrix's rows and columns. For a determinant to be even, at least one row or column must sum to an even number. Given the even distribution of digits (0, 2, 4, 6, 8 are even and 1, 3, 5, 7, 9 are odd), the probability of each row or column sum being even is incredibly high, especially in a large matrix like \(1000 \times 1000\). For the determinant to be odd, however, each row and each column must sum to an odd number, which is highly unlikely due to the high probability of having at least one row or column with an even sum.
Random Matrix Entries
Random matrix entries refer to the process of filling a matrix with values that are selected at random from a specific set, like the digits 0 to 9. In our case, each entry in a \(1000 \times 1000\) matrix is chosen independently and at random. This means that \(P(\text{even entry}) = P(\text{odd entry}) = \frac{1}{2}\). Because of this randomness and the equal likelihood, we expect a balanced mix of even and odd values across the matrix. This random distribution has significant implications on the matrix's properties, such as the determinant, making it overwhelmingly likely that the determinant will turn out to be even.

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Most popular questions from this chapter

101\. Let \(f\) be a continuous function on the real numbers. Define a sequence of functions \(f_0=f, f_1, f_2, \ldots\) by repeated integration, as follows: $$ f_0(x)=f(x) \quad \text { and } \quad f_{i+1}(x)=\int_0^x f_i(t) d t, \quad \text { for } i=0,1,2, \ldots $$ Show that for any continuous function \(f\) and any real number \(x\), $$ \lim _{n \rightarrow \infty} f_n(x)=0 . $$

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