solnsExamI+2j - SOLUTIONS EXAM I(1(20 points Answer True or...

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SOLUTIONS, EXAM I (1)(20 points) Answer True or False for each statement and give a brief reason for your answer. ( a ) If A and B are n × n matrices and AB is nonsingular then both A and B are nonsingular. True: det( AB ) = det A det B, so det AB = 0 if and only if det A = 0 and det B = 0 . Then use that A is non-singular if and only if det A = 0 . ( b ) If the reduced row echelon form of the square matrix A has all zero entries in the last row, then A is nonsingular. False: counterexample 1 0 0 0 has all zeros in last row, its determinant is 0 and is therefore singular. ( c ) det a 11 + b 11 a 12 + b 12 . . . a 1 n + b 1 n a 21 a 22 . . . a 2 n . . . . . . a n 1 a n 2 . . . a nn = det a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . a n 1 a n 2 . . . a nn + det b 11 b 12 . . . b 1 n a 21 a 22 . . . a 2 n . . . . . . a n 1 a n 2 . . . a nn 1
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2 True: perform the determinant expansion using the first row, det a 11 + b 11 a 12 + b 12 . . . a 1 n + b 1 n a 21 a 22 . . . a 2 n . . . . . . a n 1 a n 2 . . . a nn = n j =1 ( a 1 j + b 1 j ) A 1 j = n j =1 a 1 j A 1 j + n j =1 b 1 j A 1 j = det a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . a n 1 a n 2 . . . a nn + det b 11 b 12 . . . b 1 n a 21 a 22 . . . a 2 n . . .
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