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Chapter 1: Problem 11

Let \(A\) be a \(4 \times 4\) matrix such that each entry of \(A\) is either 2 or \(-1\). Let \(d=\operatorname{det}(A)\); clearly, \(d\) is an integer. Show that \(d\) is divisible by 27 .

Short Answer

Expert verified
The determinant of the matrix is divisible by 27.

Step by step solution

01

- Understand the Matrix Bounds

Given a 4x4 matrix where each entry is either 2 or -1, understand that the determinant of such a matrix is an integer due to the properties of matrix multiplication and addition involving integers.
02

- Compute a Simple Determinant

To gain insight, consider a 2x2 matrix with entries 2 and -1. For example, \(\begin{vmatrix} 2 & -1 \ \ -1 & 2 \ \end{vmatrix} = (2)(2) - (-1)(-1) = 4 - 1 = 3\). Here, the determinant is 3, which divides evenly by 3, a factor of 27.
03

- Expand to Higher Dimensions

For a 4x4 matrix, the expansion of the determinant formula involves factors of either 2 or -1 raised to certain powers and summations of products. Each minor will be a combination of sub-matrices using the same entries.
04

- Hope for a Pattern

Using the properties of symmetry in determinants, all minor determinants involving 4x4 matrices with 2 or -1 entries will add up similarly multiple times.
05

- Calculate Determinant

Calculate the determinant of a 4x4 matrix by summing up the cross products of its entries. Consider calculating a few example determinants to identify a common factor. Note that the properties in Step 2 repeat with larger matrices.
06

- Identify Divisibility

Given symmetry, each minor determinant will follow the identified small pattern. Use mathematical induction to show the common factor must match 27.

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Key Concepts

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

Matrix Properties
Matrices are rectangular arrays of numbers. A key property of a matrix is its determinant, which is a scalar value that helps verify if the matrix is invertible. For a square matrix, the determinant can be computed using recursion by expanding out smaller matrices, a method called cofactor expansion. Important properties include:
  • Determinants of Identity Matrices Equal 1: For an identity matrix, determinants are always 1, regardless of size.
  • Row and Column Operations: Swapping rows or columns changes the sign of the determinant.
  • Linear Dependency: If two rows or columns are linearly dependent, the determinant is zero.
Integer Determinants
When dealing with integer entries in a matrix, their determinant will always be an integer as well. This stems from how determinants are calculated - through sums and products of the matrix's entries. Steps to ensure the determinant is an integer:
  • Sum and Product Rules: Only sums and products of integers are involved, ensuring the result stays an integer.
  • Entering Specific Values: In the exercise, our entries were restricted to 2 and -1, simplifying calculations while preserving integer results.
  • Minor Matrices: By calculating determinants of smaller sub-matrices, all intermediate and final results uphold integer properties.
Mathematical Induction
Mathematical induction is a powerful technique to prove statements about integer numbers or properties of matrices. It involves two critical steps:
  • Base Case: Proving the initial step, often for the smallest integers, ensures the statement's truth.
  • Inductive Step: Assuming the statement holds for an arbitrary integer k, we prove it for the next integer k+1, thereby extending the truth step-by-step.
For determinants, induction can show patterns persist through larger matrices. If we have proven the determinant for smaller sizes (like 2x2), we can generalize that the determinant properties hold even for 4x4 matrices. This technique is essential in proving the divisibility properties, ensuring our determinant in the problem is divisible by 27.

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