Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
**Problem 5.** Can tic-tac-toe be completed (5 ones and 4 zeros in \(A\)) so that \(\text{rank}(A) = 2\) but neither side passed up a winning move? In this problem, we are asked to consider a tic-tac-toe game that is filled with five '1's and four '0's such that the matrix representation \(A\) has a rank of 2. Additionally, it's specified that neither player missed an opportunity to make a winning move throughout the game. This implies the game must end in a draw. The challenge is to determine if such a scenario is possible.
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Transcribed Image Text:**Problem 5.** Can tic-tac-toe be completed (5 ones and 4 zeros in \(A\)) so that \(\text{rank}(A) = 2\) but neither side passed up a winning move? In this problem, we are asked to consider a tic-tac-toe game that is filled with five '1's and four '0's such that the matrix representation \(A\) has a rank of 2. Additionally, it's specified that neither player missed an opportunity to make a winning move throughout the game. This implies the game must end in a draw. The challenge is to determine if such a scenario is possible.
Expert Solution
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Step 1: Introduction

To answer this question, we need to understand what the rank of a matrix represents in the context of tic-tac-toe.

In tic-tac-toe, we can represent the game board as a 3 cross times 3 matrix A, where each entry is either a 1 (representing a move by player X) or a 0 (representing a move by player O). The rank of this matrix represents the dimension of the subspace of the vector space spanned by the rows (or columns) of the matrix. In other words, the rank of A is the maximum number of rows (or columns) that are linearly independent.

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