quiz_2_solution - i.e a ik = a jk for all k = 1 2,n 1 Now...

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Solutions of Quiz 2 5. Use mathematical induction to prove that if A is an ( n + 1) × ( n + 1) matrix with two identical rows then det( A )=0. Proof. Use induction on n . n = 2, then it is clear that det( A )=0. Suppose the statement is hold for A which is a n × n matrix. If A is a ( n + 1) × ( n + 1) matrix with two identical rows, show that det( A )=0. Suppose that the i -th row and j -th row are identical.
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Unformatted text preview: i.e a ik = a jk for all k = 1 , 2 ,...,n + 1. Now we compute det( A ) by expand by the r-th row of A with r 6 = i or j . ⇒ det( A ) = n +1 ∑ k =1 (-1) r + k a rk det( M rk ) Note that M rk are n × n matrix and M rk has two identical rows. By induction hypothesis, det M rk =0 for k = 1 , 2 ,...,n + 1. ⇒ det(A) = 0 ....
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