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.

Answer: For EA we have

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.

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