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no—an invertible matrix has exactly one inverse.
THEOREM 1.4.4
If B and C are both inverses of the matrix A, then .
Proof Since B is an inverse of A, we have . Multiplying both sides on the right by C gives . But
, so .
As a consequence of this important result, we can now speak of “the” inverse of an invertible matrix. If A is invertible, then its
inverse will be denoted by the symbol . Thus,
The inverse of A plays much the same role in matrix arithmetic that the reciprocal plays in the numerical relationships
and .
In the next section we shall develop a method for finding inverses of invertible matrices of any size; however, the following
theorem gives conditions under which a matrix is invertible and provides a simple formula for the inverse.
THEOREM 1.4.5
The matrix
is invertible if , in which case the inverse is given by the formula
Proof We leave it for the reader to verify that and .
THEOREM 1.4.6
If A and B are invertible matrices of the same size, then is invertible and
