Let A = [ [0, 1], [1, 0] ]. Which of the following is a correct statement about A^2 and A^3?

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Multiple Choice

Let A = [ [0, 1], [1, 0] ]. Which of the following is a correct statement about A^2 and A^3?

This question tests how powers of a swap matrix behave. The matrix A swaps the two coordinates, so doing the swap twice undoes it, giving the identity. Compute A squared: multiplying A by itself gives [[1,0],[0,1]] = I. Then A cubed is A^2 times A, which is I times A, so A^3 = A. So the square is the identity and the cube returns the original swap matrix. This also shows a pattern: even powers give I, odd powers give A. The other possibilities would imply A^2 = A or A^3 = -A, which contradicts the actual results A^2 = I and A^3 = A.

This question tests how powers of a swap matrix behave. The matrix A swaps the two coordinates, so doing the swap twice undoes it, giving the identity.

Compute A squared: multiplying A by itself gives [[1,0],[0,1]] = I. Then A cubed is A^2 times A, which is I times A, so A^3 = A. So the square is the identity and the cube returns the original swap matrix.

This also shows a pattern: even powers give I, odd powers give A. The other possibilities would imply A^2 = A or A^3 = -A, which contradicts the actual results A^2 = I and A^3 = A.

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