college linear algebra

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Label the following statements as True or False. If the statement is False, give a counterexample. (a) If two rows of a matrix A are identical, then det(A) = 0. (b) If B is a matrix obtained from a square matrix A by multiplying a row of A by a scalar, then det(B) = det(A). (c) If B is a matrix obtained from a square matrix A by interchanging any two rows, then det(B) = −det(A). (d) If B is a matrix obtained from a square matrix A by adding k times row i to row j, then det(B) = kdet(A). (e) If A ∈ Mn×n(R) has rank n, then det(A) = 0.

3. (10 points/5 each) Use the permutation formula from the end of lecture 15 to compute the determinants of the following matrices. Then, use the pivot formula to verify your result. Clearly show all solution steps.

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