Math Problem Statement

Let A be a square matrix that is filled with all zeros except for the coordinates where the row number equals the column number. In those cells, the numbers from 1 to n appear in alphabetical order based on each number’s English spelling. For example if n=3 then the order would be 1-3-2. Find the trace of A**2

Solution

To solve this problem, we first need to understand what the matrix AA looks like and then compute the trace of A2A^2.

Step 1: Understanding the Matrix AA

The matrix AA is an n×nn \times n square matrix. It is a diagonal matrix where the diagonal elements are arranged in alphabetical order based on the English spelling of numbers from 1 to nn. For example:

  • If n=3n = 3, the numbers in alphabetical order would be: 1 (One), 3 (Three), 2 (Two).
  • This gives us the diagonal matrix: 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \\ \end{pmatrix}$$

Step 2: Finding A2A^2

Since AA is a diagonal matrix, squaring it is straightforward. The diagonal elements of A2A^2 will simply be the squares of the diagonal elements of AA. For the example with n=3n = 3:

1^2 & 0 & 0 \\ 0 & 3^2 & 0 \\ 0 & 0 & 2^2 \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 4 \\ \end{pmatrix}$$ ### Step 3: Finding the Trace of $$A^2$$ The trace of a matrix is the sum of the diagonal elements. In this case, the trace of $$A^2$$ is: $$\text{Trace}(A^2) = 1 + 9 + 4 = 14$$ ### General Case for $$n$$ To generalize for any $$n$$, we need to compute the trace of $$A^2$$, which will be the sum of the squares of the numbers in the diagonal of $$A$$, listed in alphabetical order. For example, if $$n = 5$$, the numbers in alphabetical order would be **1 (One), 4 (Four), 5 (Five), 3 (Three), 2 (Two)**. Then the matrix $$A$$ would be: $$A = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 & 0 \\ 0 & 0 & 5 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 2 \\ \end{pmatrix}$$ And $$A^2$$ would be: $$A^2 = \begin{pmatrix} 1^2 & 0 & 0 & 0 & 0 \\ 0 & 4^2 & 0 & 0 & 0 \\ 0 & 0 & 5^2 & 0 & 0 \\ 0 & 0 & 0 & 3^2 & 0 \\ 0 & 0 & 0 & 0 & 2^2 \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 16 & 0 & 0 & 0 \\ 0 & 0 & 25 & 0 & 0 \\ 0 & 0 & 0 & 9 & 0 \\ 0 & 0 & 0 & 0 & 4 \\ \end{pmatrix}$$ The trace of $$A^2$$ would be: $$\text{Trace}(A^2) = 1 + 16 + 25 + 9 + 4 = 55$$ ### Conclusion The trace of $$A^2$$ is the sum of the squares of the numbers in alphabetical order. For the case when $$n = 3$$, the trace is 14. Would you like further clarification or more examples? --- ### Five Related Questions: 1. How do you determine the alphabetical order of numbers for any $$n$$? 2. How do you compute the trace of a general square matrix? 3. What are the properties of diagonal matrices in matrix multiplication? 4. How is matrix squaring related to diagonalization? 5. What is the significance of the trace of a matrix in linear algebra? **Tip:** Diagonal matrices are much easier to work with in operations like squaring or finding the trace, since only the diagonal elements matter!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Operations
Diagonal Matrices
Trace of a Matrix

Formulas

Trace of A^2 = Sum of squares of diagonal elements
Matrix squaring for diagonal matrices

Theorems

Properties of Diagonal Matrices
Matrix Trace Theorem

Suitable Grade Level

Grades 11-12 or Undergraduate