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Recall the definition of linear independence. The columns of X are said to be linearly...

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haqc | (Level 1) Honors

Posted October 3, 2013 at 9:27 PM via web

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  1. Recall the definition of linear independence. The columns of X are said to be linearly depen- dent if there exists a p×1 vector v 0 with Xv = 0. We will say that the columns of X are linearly independent if Xv = 0 implies v = 0. Let A be a square matrix. Show that if the columns of A are linearly dependent, A−1 cannot exist. Hint: v cannot be both zero and not zero at the same time. 

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aruv | High School Teacher | (Level 2) Valedictorian

Posted October 4, 2013 at 4:26 AM (Answer #1)

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Let A be a square matrix. Let order of matrix A be n. The columns of A are linearly dependents then rank(A) will less than n. Matrix A is invertible if order of A is same as rank of A i.e. order(A)=rank(A). In case columns are linearly dependents  order(A)>rank(A). Therefore matrix A will noninvertble. It means `A^(-1)` does not exist.

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