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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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