Answer to Question #345562 in Linear Algebra for Zee

Question #345562

Find an expression for a square matrix A satisfying A squared

= In, where In is the n × n identity matrix. Give 3

examples for the case n = 3. 


1
Expert's answer
2022-06-01T03:33:07-0400

Matrix "A" is an involutory matrix, if "A^2=I_n" .

There is no formula for this type matrix, but there many examples.

All matrix "A=\\begin{pmatrix}\n\\pm 1 &0&0&\u2026&0\n\\\\\n0& \\pm1&0&\u2026&0\n\\\\\n0&0&\\pm1&\u2026&0\n\\\\\n\\vdots &\\vdots &\\vdots& \\ddots &0\n\\\\\n0&0&0&0&\\pm1\n\\end{pmatrix}" satisfy "A^2=I_n" .


Examples of "3\\times 3" matrices:

"A=\\begin{pmatrix}\n1&0&0\\\\0&1&0\\\\0&0&1\n\\end{pmatrix}"

"B=\\begin{pmatrix}\n-1&0&0\\\\0&1&0\\\\0&0&-1\n\\end{pmatrix}"

"C=\\begin{pmatrix}\n1&0&0\\\\0&0&-1\\\\0&-1&0\n\\end{pmatrix}"


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