Linear Algebra and Its Applications, Exercise 1.5.16

Exercise 1.5.16. Find a 4 by 4 matrix (preferably a permutation matrix) that is nonsingular and for which elimination requires three row exchanges.

Answer: The following permutation matrix meets the requirement:

\begin{bmatrix} 0&0&0&1 \\ 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \end{bmatrix}

For this matrix elimination requires the following row exchanges:

  1. Exchange row 1 and row 2.
  2. Exchange row 2 and row 3.
  3. Exchange row 3 and row 4.

After the row exchanges the matrix is the identity matrix, which is nonsingular:

\begin{bmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix}

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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