## Linear Algebra and Its Applications, Review Exercise 1.14

Review exercise 1.14. For each of the following find a 3 by 3 matrix $B$ such that for any matrix $A$

(a) $BA = 2A$

(b) $BA = 2B$

(c) The first row of $BA$ is the last row of $A$ and the last row of $BA$ is the first row of $A$.

(d) The first column of $BA$ is the last column of $A$ and the last column of $BA$ is the first column of $A$.

Answer: (a) We have $2A = 2IA$ so that we can choose $B = 2I$ $B = \begin{bmatrix} 2&0&0 \\ 0&2&0 \\ 0&0&2 \end{bmatrix}$

(b) For all $A$ we have $0 \cdot A = 0 = 2 \cdot 0$ so that we can choose $B = 0$ $B = \begin{bmatrix} 0&0&0 \\ 0&0&0 \\ 0&0&0 \end{bmatrix}$

(c) We can choose $B$ to be the permutation matrix that reverses the order of rows in $A$ $B = \begin{bmatrix} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{bmatrix}$

For example $BA = \begin{bmatrix} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{bmatrix} \begin{bmatrix} 1&2&3 \\ 4&5&6 \\ 7&8&9 \end{bmatrix} = \begin{bmatrix} 7&8&9 \\ 4&5&6 \\ 1&2&3 \end{bmatrix}$

(d) The first column of $BA$ is produced by multiplying the first row of $B$ by the first column of $A$. Since this computation does not involve the last column of $A$ in general it is impossible to find a matrix $B$ such that the first column of $BA$ is equal to the last column of $A$.

However note that we can reverse the order of columns in $A$ by multiplying $A$ on the right by the value of $B$ from (c) above: $AB = \begin{bmatrix} 1&2&3 \\ 4&5&6 \\ 7&8&9 \end{bmatrix} \begin{bmatrix} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{bmatrix} = \begin{bmatrix} 3&2&1 \\ 6&5&4 \\ 9&8&7 \end{bmatrix}$

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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