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Even though one irreducible quadratic gives two

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Unformatted text preview: 1 to be same dimensions as A, that is, n × n. Since not all matrices are square, not all matrices are invertible. However, just because a matrix is square doesn’t guarantee it is invertible. (See the exercises.) Our first result summarizes some of the important characteristics of invertible matrices and their inverses. Theorem 8.6. Suppose A is an n × n matrix. 1. If A is invertible then A−1 is unique. 2. A is invertible if and only if AX = B has a unique solution for every n × r matrix B . The proofs of the properties in Theorem 8.6 rely on a healthy mix of definition and matrix arithmetic. To establish the first property, we assume that A is invertible and suppose the matrices B and C act as inverses for A. That is, BA = AB = In and CA = AC = In . We need to show that B and C are, in fact, the same matrix. To see this, we note that B = In B = (CA)B = C (AB ) = CIn = C . Hence, any two matrices that act like A−1 are, in fact, the same matrix.4 To prove the second property of Theorem 8.6, we note that if A is invertible then the d...
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