## Linear Algebra and Its Applications, Exercise 1.6.7

Exercise 1.6.7. Find three 2 by 2 matrices A such that $A^2 = I$

and A is neither I nor -I.

Answer: We first note that the transpose of I is its own inverse: $\begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix} \begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix} = \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}$

Note that this also follows from the result of exercise 1.6.2 that $PP^T = I$

where P is a permutation matrix, since the 2 by 2 matrix above is a permutation of I.

Using trial and error we can find a second such matrix: $\begin{bmatrix} 2&-1 \\ 3&-2 \end{bmatrix} \begin{bmatrix} 2&-1 \\ 3&-2 \end{bmatrix} = \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}$

and a third: $\begin{bmatrix} \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}} \end{bmatrix}= \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}$

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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