Explain why the columns of A^2 span R^n whenever the columns of an n X n matrix

Question

Explain why the columns of A^2 span R^n whenever the columns of an n X n matrix A are linearly independent. Choose the correct answer below. Note that the invertible matrix theorem is abbreviated IMT. A n If the columns of A are linearly independent, then it directly follows that the columns of A^2 span R^n B. If the columns of A are linearly independent and A is square, then A is invertible, by the IMT. Thus, A^2, which is the product of invertible matrices, is also invertible. So, by the IMT, the columns of A^2 span R^n. C. If the columns of A are linearly independent and A is square, then A is invertible, by the IMT. Thus, A^2, which is the product of invertible matrices, is not invertible. So, the columns of A^2 span R^n. D. If the columns of A are linearly independent and A is square, then A is not invertible. Thus, A^2, which is the product of non invertible matrices, is also not invertible. So, the columns of A^2 span R^n

Explanation / Answer

Answer: B

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