Linear Algebra and Its Applications, Review Exercise 1.1

Review exercise 1.1. (a) Show the 3 by 3 matrices $A$ and $B$ for which $a_{ij} = i - j$ and $b_{ij} = i/j$ .

(b) Find the products $AB$ , $BA$ , and $A^2$ of the above matrices.

Answer: (a) We have

$A = \begin{bmatrix} 1-1&1-2&1-3 \\ 2-1&2-2&2-3 \\ 3-1&3-2&3-3 \end{bmatrix} = \begin{bmatrix} 0&-1&-2 \\ 1&0&-1 \\ 2&1&0 \end{bmatrix}$

and

$B = \begin{bmatrix} 1/1&1/2&1/3 \\ 2/1&2/2&2/3 \\ 3/1&3/2&3/3 \end{bmatrix} = \begin{bmatrix} 1&\frac{1}{2}&\frac{1}{3} \\ 2&1&\frac{2}{3} \\ 3&\frac{3}{2}&1 \end{bmatrix}$

(b) We have

$AB = \begin{bmatrix} 0&-1&-2 \\ 1&0&-1 \\ 2&1&0 \end{bmatrix} \begin{bmatrix} 1&\frac{1}{2}&\frac{1}{3} \\ 2&1&\frac{2}{3} \\ 3&\frac{3}{2}&1 \end{bmatrix}$

$= \begin{bmatrix} 0-2-6&0-1-3&0-\frac{2}{3}-2 \\ 1+0-3&\frac{1}{2}+0-\frac{3}{2}&\frac{1}{3}+0-1 \\ 2+2+0&1+1+0&\frac{2}{3}+\frac{2}{3}+0 \end{bmatrix} = \begin{bmatrix} 8&-4&-\frac{8}{3} \\ -2&-1&-\frac{2}{3} \\ 4&2&\frac{4}{3} \end{bmatrix}$

$BA = \begin{bmatrix} 1&\frac{1}{2}&\frac{1}{3} \\ 2&1&\frac{2}{3} \\ 3&\frac{3}{2}&1 \end{bmatrix} \begin{bmatrix} 0&-1&-2 \\ 1&0&-1 \\ 2&1&0 \end{bmatrix}$

$= \begin{bmatrix} 0+\frac{1}{2}+\frac{2}{3}&-1+0+\frac{1}{3}&-2-\frac{1}{2}+0 \\ 0+1+\frac{4}{3}&-2+0+\frac{2}{3}&-4-1+0 \\ 0+\frac{3}{2}+2&-3+0+1&-6-\frac{3}{2}+0 \end{bmatrix} = \begin{bmatrix} \frac{7}{6}&-\frac{2}{3}&-\frac{5}{2} \\ \frac{7}{3}&-\frac{4}{3}&-5 \\ \frac{7}{2}&-2&-\frac{15}{2} \end{bmatrix}$

$A^2 = \begin{bmatrix} 0&-1&-2 \\ 1&0&-1 \\ 2&1&0 \end{bmatrix} \begin{bmatrix} 0&-1&-2 \\ 1&0&-1 \\ 2&1&0 \end{bmatrix}$

$= \begin{bmatrix} 0-1-4&0+0-2&0+1+0 \\ 0+0-2&-1+0-1&-2+0+0 \\ 0+1+0&-2+0+0&-4-1+0 \end{bmatrix} = \begin{bmatrix} -5&-2&1 \\ -2&-2&-2 \\ 1&-2&-5 \end{bmatrix}$

(As a point of interest, note that $A = -A^T$ and $A^2 = (A^2)^T$ . This follows from equation 1M(i) in section 1.6: $(A^2)^T = (AA)^T = A^TA^T = (-A)(-A) = AA = A^2$ .)

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.

2 Responses to Linear Algebra and Its Applications, Review Exercise 1.1

Victor says:

September 23, 2013 at 1:08 am

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- hecker says:
  
  September 23, 2013 at 3:30 am
  
  I’m glad you like the site and find it useful. (If you have a blog please feel free to link to the site.) Good luck studying linear algebra!