Linear Algebra and Its Applications, Review Exercise 1.21

Review exercise 1.21. Given the 2 by 2 matrix

$D = \begin{bmatrix} 2&0 \\ 0&5 \end{bmatrix}$

describe the rows of $DA$ and the columns of $AD$ .

Answer: When multiplying $A$ from the left by $D$ to produce $DA$ the first row of $DA$ will be 2 times the first row of $A$ and the second row of $DA$ will be 5 times the second row of $A$ . For example

$\begin{bmatrix} 2&0 \\ 0&5 \end{bmatrix} \begin{bmatrix} 1&2 \\ 3&4 \end{bmatrix} = \begin{bmatrix} 2&4 \\ 15&20 \end{bmatrix}$

When multiplying $A$ from the right by $D$ to produce $AD$ the first column of $AD$ will be 2 times the first column of $A$ and the second column of $AD$ will be 5 times the second column of $A$ . For example

$\begin{bmatrix} 1&2 \\ 3&4 \end{bmatrix} \begin{bmatrix} 2&0 \\ 0&5 \end{bmatrix} = \begin{bmatrix} 2&10 \\ 6&20 \end{bmatrix}$

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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