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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