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.



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Posted on (Answer #1)

Let A is the square matrix of order n. If column of matrix A is linearly dependent then rank of the matrix A will be less than n. But we  know matrix is invertible if rank of the matrix equal to the order of the matrix i.e.



`rank(A)<n=order (A)`

Therefore matrix A is non invertble i.e. `A^(-1)` does not exist.

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