Linear Algebra and Its Applications, Review Exercise 1.8

Review exercise 1.8. Given the following matrices:

E = \begin{bmatrix} 1&0&0 \\ 0&2&0 \\ 4&0&1 \end{bmatrix} \quad \rm or \quad E = \begin{bmatrix} 1&1&1 \\ 0&0&0 \end{bmatrix} \quad \rm or \quad E = \begin{bmatrix} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{bmatrix}

for a matrix A how are the rows of EA related to the rows of A?

Answer: For the first matrix E, the product EA is a 3 by 3 matrix in which:

  1. The first row is equal to the first row of A.
  2. The second row is equal to the second row of A multiplied by 2.
  3. The third row is equal to the sum of 4 times the first row of A and the third row of A.

For the second matrix E, the product matrix EA is a 2 by 3 matrix in which:

  1. The first row is equal to the sum of all three rows of A.
  2. The second row is equal to zero.

The third matrix E is a permutation matrix that reverses the order of rows, so that in the product EA:

  1. The first row is equal to the third row of A.
  2. The second row is equal to the second row of A.
  3. The third row is equal to the first row of A.

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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