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Is it possible for a 4 by 4 matrix to be invertible when its columns do not span R4?

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No, it is not possible for a 4 by 4 matrix to be invertible when its columns do not span R4. If the columns of a 4 by 4 matrix do not span R4, then the matrix is not full rank, which means that its rank is less than 4. By the invertible matrix theorem, a square matrix is invertible if and only if its rank is equal to its size, which in this case is 4. Since a matrix with linearly dependent columns has rank less than 4, it follows that it is not invertible. Therefore, a 4 by 4 matrix cannot be invertible when its columns do not span R4.
User Ankita Kuchhadiya
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