## Linear Algebra and Its Applications, exercise 1.4.15

Exercise 1.4.15. Given the matrix E given by $E = \begin{bmatrix} 1&7 \\ 0&1 \end{bmatrix}$

and an arbitrary 2×2 matrix A, describe the rows of the product matrix EA and the columns of AE. $EA = \begin{bmatrix} 1&7 \\ 0&1 \end{bmatrix} \begin{bmatrix} a_{11}&a_{12} \\ a_{21}&a_{22} \end{bmatrix}$

The first row of EA is a linear combination of the two rows of A, with coefficients 1 and 7 respectively; in other words, the first row of EA is equal to the first row of A plus 7 times the second row of A. The second row of EA is equal to the second row of A.

For AE we have $AE = \begin{bmatrix} a_{11}&a_{12} \\ a_{21}&a_{22} \end{bmatrix} \begin{bmatrix} 1&7 \\ 0&1 \end{bmatrix}$

The first column of AE is equal to the first column of A. The second column of AE is a linear combination of the two columns of A, with coefficients 7 and 1 respectively; in other words, the second column of AE is equal to 7 times the first column of A plus the second column 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 .

This entry was posted in linear algebra. Bookmark the permalink.