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

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 .

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

1. Victor says:

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• hecker says:

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