The invertible matrix theorem is a theorem in linear algebra which gives a series of equivalent conditions for an n×n square matrix A to have an inverse. In particular, A is invertible if and only if any (and hence, all) of the following hold: 1.A is row-equivalent to the n×n identity matrix I_n. 2.A has n pivot positions. 3. The equation A x = 0 has only the trivial solution x = 0. 4. The columns of A form a linearly independent set. 5. The linear transformation x↦A x is one-to-one. 6. For each column vector b element R^n, the equation A x = b has a unique solution.
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