The Invertible Matrix Theorem

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( 1 ⇐⇒ 2 ) : The matrix A has n pivots if and only if its reduced row echelon form is the identity matrix I n . This happens exactly when the procedure in Section 3.5 to compute the inverse succeeds.

( 2 ⇐⇒ 3 ) : The null space of a matrix is { 0 } if and only if the matrix has no free variables, which means that every column is a pivot column, which means A has n pivots. See this recipe in Section 2.6.

( 2 ⇐⇒ 4,2 ⇐⇒ 5 ) : These follow from this recipe in Section 2.5 and this theorem in Section 2.3, respectively, since A has n pivots if and only if has a pivot in every row/column.

( 4 + 5 ⇐⇒ 6 ) : We know Ax = b has at least one solution for every b if and only if the columns of A span R n by this theorem in Section 3.2, and Ax = b has at most one solution for every b if and only if the columns of A are linearly independent by this theorem in Section 3.2. Hence Ax = b has exactly one solution for every b if and only if its columns are linearly independent and span R n .

( 1 ⇐⇒ 7 ) : This is the content of this theorem in Section 3.5.

( 7 = ⇒ 8 + 9 ) : See this proposition in Section 3.5.

( 8 ⇐⇒ 4,9 ⇐⇒ 5 ) : See this this theorem in Section 3.2 and this theorem in Section 3.2.

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