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.

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

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

  1. Victor says:

    This is my first year studying linear algebra, I’m studying from Strang’s textbook and only by luck I found this website.
    I think that this is amazing, more people should help you out.
    Thanks for this great work.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s