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  3. determine if the following statements are true always or false...

Question: determine if the following statements are true always or false...

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Determine if the following statements are true (always) or false. If true, give a proof. If false, give a counterexample.

(a) If the product AB is a square matrix, then the product BA is defined.

(b) If A and B are matrices of the same size, then AB is defined.

(c) If the products AB and BA are defined, then AB and BA are square matrices.

(d) If A is a square matrix and A != O and A != I, then A^2 != A.

(e) Matrix multiplication is commutative.

(f) If A and B are square matrices of the same size, then (A + B)(A − B) = A^2 − B^2 .

(g) If A and B are square matrices of the same size, then (A + B) ^2 = A^2 + 2AB + B^2 .

(h) If A^2 = A, then A^n = A for all positive integer values of n.

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