Page 74 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Invertibility of a Transpose

The following theorem establishes a relationship between the inverse of an invertible matrix and the inverse of its transpose.
THEOREM 1.4.10

If A is an invertible matrix, then is also invertible and
(4)

Proof We can prove the invertibility of and obtain 4 by showing that

But from part (d) of Theorem 1.4.9 and the fact that , we have

which completes the proof.

EXAMPLE 10 Verifying Theorem 1.4.10
Consider the matrices

Applying Theorem 1.4.5 yields

As guaranteed by Theorem 1.4.10, these matrices satisfy 4.

Exercise Set 1.4

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Let
1.