## 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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