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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One Response to Linear Algebra and Its Applications, Review Exercise 1.21

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