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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aruv | High School Teacher | (Level 2) Valedictorian

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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)=order(A)=n`  

But

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

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

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