Question: 2. Let A, B, C, and D be (n x n)-matrices with A invertible. (i) Show that det A C D = det(A) det(D). Hint:

2. Let A, B, C, and D be (n x n)-matrices with A invertible. (i) Show that det A C D = det(A) det(D). Hint: (ii) Find matrices X and Y which produce the block L U factorization (iii) Show that det | # S] ) = det(A) det(D - BA-' C). (iv) If AB = BA then prove that det ( A S]) = det(AD - BC). (v) If AB # BA, then give an example such that det( S ) * det(AD - BC)

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