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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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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