Linear Algebra and Its Applications, Exercise 1.6.5

Exercise 1.6.5. For a matrix A assume that $A^2$ is invertible and has inverse B. Prove that A is also invertible, with inverse AB.

Answer: We have

$(A^2)^{-1} = B \rightarrow A^2= B^{-1}$

We then have

$A(AB) = A^2B = B^{-1}B = I$

We also have

$(AB)A = BB^{-1}(AB)A = BA^2(AB)A = BA(A^2)BA$

$= BAB^{-1}BA = BAIA = BA^2 = BB^{-1} = I$

So AB is both a left and right inverse of A, and thus

$A^{-1} = AB$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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