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Unformatted text preview: 5.19 Let A be an 3 3 matrix, Suppose the third row minus the rst row equals the second row. Explain why A cannot be invertible. SOLUTION Let B = A t , and write B = [ v 1 , v 2 , v 3 ]. Since, by the denition of the transpose, the columns of B are the rows of A , v 3-v 1 = v 2 , That is, v 1 + v 2-v 3 = 0. Then by V.I.F. 1 , B 1 1-1 = 0 . Therefore, by the bullet point containing (3.20), B = A t has no left inverse. Now if A has a right inverse C , then AC = I , so C t A t = I t = I , and so C t would be a left inverse of A t . But we have just seen that this is impossible. Hence A has no right inverse. It is therefore not invertible. 21 /september/ 2005; 22:09 41...
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- Fall '08