Final answer:
To find integer matrices A and B such that A² = B² = I and AB = BA, we can use the concept of eigenvectors and eigenvalues.
Step-by-step explanation:
To find integer matrices A and B such that A² = B² = I and AB = BA, we can use the concept of eigenvectors and eigenvalues. Let's assume that A and B are 2x2 matrices. Since A² = B² = I, the eigenvalues of A and B must be 1 and -1. The eigenvectors corresponding to the eigenvalue 1 can be any non-zero vector; let's choose (1, 0). The eigenvectors corresponding to the eigenvalue -1 can be any vector orthogonal to (1, 0); let's choose (0, 1). Therefore, the matrix A can be [1 0; 0 -1] and the matrix B can be [1 0; 0 1]. These matrices satisfy A² = B² = I and AB = BA.