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Inverses and Elementary Matrices

Part 5: Summary

  1. How can row operations help you determine whether a square matrix is invertible?

  2. How can row operations help you find the inverse of an invertible square matrix?

  3. How is the inverse of a product AB related to the inverses of A and B?

  4. How is the inverse of the transpose of A related to the inverse of A?

  5. How rare are invertible matrices? If you construct a random square matrix, what is the probability that it will be invertible?

  6. Explain why every invertible matrix is a product of elementary matrices.

  7. If A and B are row equivalent matrices, explain why there must be an invertible matrix P such that PA = B.

  8. (optional) What is a Hilbert matrix? What special characteristic do Hilbert matrices have? What is the signficance of that characteristic for finding inverse matrices?
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