(adsbygoogle = window.adsbygoogle || []).push({}); The problem statement, all variables and given/known data

Let M be an n x n matrix. Prove that the columns of M form an orthonormal set if and only if M^{-1}= M^{T}.

The attempt at a solution

Let's consider the following 2 x 2 matrix

[tex]\begin{pmatrix} a & b \\ c & d \end{pmatrix}[/tex]

If the inverse equals its transpose, i.e.

[tex]\frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} a & c \\ b & d \end{pmatrix}[/tex]

then

[tex]\begin{align*}

a & = \frac{d}{ad - bc} \\

b & = \frac{-c}{ad - bc} \\

c & = \frac{-b}{ad - bc} \\

d & = \frac{a}{ad - bc}

\end{align*}[/tex]

or rather that ad - bc = 1. Does this mean that ab + cd = 0? Not necessarily: Consider a = 2 and b = c = d = 1 so that ad - bc = 2 - 1 = 1 but ab + cd = 2 + 1 = 3. What gives?

And doeas ab + cd = 0 imply the equations above for a, b, c and d? I think not.

I'm stumped.

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# Homework Help: A Matrix with Orthonormal Columns

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