# Determine if the statement is true or false, and justify your answer. If A and B are diagonalizable

Determine if the statement is true or false, and justify your answer.

If A and B are diagonalizable n x nm matrices, then so is AB.

© True. Since the determinant of the product of matrices equals the product of the determinants of the matrices, AB is diagonalizable if and only if A and 8 are diagonalizable.

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© False. Consider A = FE 3 | and B 10 |

O True. Since A = PD,P~! and B = PD2P~!, AB = PD,P~1PD2P-! = PD,D2P™. Since the product of diagonal matrices is diagonal, AB is diagonalizable.

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© False. Consider A = FE | and B 10 |

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© False. Consider A FE 3 | and B [2 1 |:

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