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Linear Algebra : Orthogonal Matrices

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Example Question #1 : Orthogonal Matrices

Determine if the following matrix is orthogonal or not.

$A=\begin{bmatrix} -1& 0\\ 0& 1 \end{bmatrix}$ $\displaystyle A=\begin{bmatrix} -1& 0\\ 0& 1 \end{bmatrix}$

Possible Answers:

$\displaystyle A$ is an orthogonal matrix

$\displaystyle A$ is not an orthogonal matrix

Correct answer:

$\displaystyle A$ is an orthogonal matrix

Explanation:

To determine if a matrix is orthogonal, we need to multiply the matrix by it's transpose, and see if we get the identity matrix.

$A=\begin{bmatrix} -1& 0\\ 0& 1 \end{bmatrix}$ $\displaystyle A=\begin{bmatrix} -1& 0\\ 0& 1 \end{bmatrix}$, $A^T=\begin{bmatrix} -1& 0\\ 0& 1 \end{bmatrix}$ $\displaystyle A^T=\begin{bmatrix} -1& 0\\ 0& 1 \end{bmatrix}$

$A.A^T=\begin{bmatrix} (-1)(-1)&(0)(0) \\ (0)(0)& (1)(1) \end{bmatrix}=\begin{bmatrix} 1&0\\ 0& 1 \end{bmatrix}$ $\displaystyle A.A^T=\begin{bmatrix} (-1)(-1)&(0)(0) \\ (0)(0)& (1)(1) \end{bmatrix}=\begin{bmatrix} 1&0\\ 0& 1 \end{bmatrix}$

Since we get the identity matrix, then we know that $\displaystyle A$ is an orthogonal matrix.

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Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}-0.60587&0.79556\\0.79556&0.60587\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}-0.60587&0.79556\\0.79556&0.60587\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$

Possible Answers:

$\begin{align*}&\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}&\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}&\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.60587&0.79556\\0.79556&0.60587\end{bmatrix}= \\&\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.60587&0.79556\\0.79556&0.60587\end{bmatrix}= \\&\text{The matrix is orthogonal.}\end{align*}$

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Example Question #342 : Linear Algebra

$\begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}0.68567&0.12975&-0.71626\\0.14807&0.93855&0.31176\\0.71269&-0.31982&0.62433\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}0.68567&0.12975&-0.71626\\0.14807&0.93855&0.31176\\0.71269&-0.31982&0.62433\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}0.68567&0.12975&-0.71626\\0.14807&0.93855&0.31176\\0.71269&-0.31982&0.62433\end{bmatrix}=1\\&\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}0.68567&0.12975&-0.71626\\0.14807&0.93855&0.31176\\0.71269&-0.31982&0.62433\end{bmatrix}=1\\&\text{The matrix is orthogonal.}\end{align*}$

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Example Question #3 : Orthogonal Matrices

$\begin{align*}&\text{Is }A=\begin{bmatrix}-0.85616&0.46933\\0.46933&0.96236\end{bmatrix}\\&\text{orthogonal?}\end{align*}$ $\displaystyle \begin{align*}&\text{Is }A=\begin{bmatrix}-0.85616&0.46933\\0.46933&0.96236\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.85616&0.46933\\0.46933&0.96236\end{bmatrix}=-1.0442\\&\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.85616&0.46933\\0.46933&0.96236\end{bmatrix}=-1.0442\\&\text{The matrix is not orthogonal.}\end{align*}$

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Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}-0.57605&-0.81742\\-0.81742&0.57605\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}-0.57605&-0.81742\\-0.81742&0.57605\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.57605&-0.81742\\-0.81742&0.57605\end{bmatrix}=-1\\&\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.57605&-0.81742\\-0.81742&0.57605\end{bmatrix}=-1\\&\text{The matrix is orthogonal.}\end{align*}$

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Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Is the matrix }A=\begin{bmatrix}-0.44125&-0.89739\\-0.89739&0.44125\end{bmatrix}\\&\text{orthogonal?}\end{align*}$ $\displaystyle \begin{align*}&\text{Is the matrix }A=\begin{bmatrix}-0.44125&-0.89739\\-0.89739&0.44125\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.44125&-0.89739\\-0.89739&0.44125\end{bmatrix}=-1\\&\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.44125&-0.89739\\-0.89739&0.44125\end{bmatrix}=-1\\&\text{The matrix is orthogonal.}\end{align*}$

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Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Is the matrix }A=\begin{bmatrix}-0.62126&-0.0069882&0.77059\\-0.86406&-0.14727&-0.61555\\0.011558&-1.0415&0.043005\end{bmatrix}\\&\text{orthogonal?}\end{align*}$ $\displaystyle \begin{align*}&\text{Is the matrix }A=\begin{bmatrix}-0.62126&-0.0069882&0.77059\\-0.86406&-0.14727&-0.61555\\0.011558&-1.0415&0.043005\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.62126&-0.0069882&0.77059\\-0.86406&-0.14727&-0.61555\\0.011558&-1.0415&0.043005\end{bmatrix}=1.0968\\&\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.62126&-0.0069882&0.77059\\-0.86406&-0.14727&-0.61555\\0.011558&-1.0415&0.043005\end{bmatrix}=1.0968\\&\text{The matrix is not orthogonal.}\end{align*}$

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Example Question #2 : Orthogonal Matrices

$\begin{align*}&\text{Is the matrix }A=\begin{bmatrix}0.012505&-0.9358&-0.11296\\-0.84627&0.011244&0.49169\\-0.3916&0.21287&-0.83217\end{bmatrix}\\&\text{orthogonal?}\end{align*}$ $\displaystyle \begin{align*}&\text{Is the matrix }A=\begin{bmatrix}0.012505&-0.9358&-0.11296\\-0.84627&0.011244&0.49169\\-0.3916&0.21287&-0.83217\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}0.012505&-0.9358&-0.11296\\-0.84627&0.011244&0.49169\\-0.3916&0.21287&-0.83217\end{bmatrix}=0.85765\\&\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}0.012505&-0.9358&-0.11296\\-0.84627&0.011244&0.49169\\-0.3916&0.21287&-0.83217\end{bmatrix}=0.85765\\&\text{The matrix is not orthogonal.}\end{align*}$

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Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Is }A=\begin{bmatrix}-0.55978&-0.80992&0.17514\\0.55713&-0.52432&-0.64396\\0.61339&-0.2629&0.74474\end{bmatrix}\\&\text{orthogonal?}\end{align*}$ $\displaystyle \begin{align*}&\text{Is }A=\begin{bmatrix}-0.55978&-0.80992&0.17514\\0.55713&-0.52432&-0.64396\\0.61339&-0.2629&0.74474\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.55978&-0.80992&0.17514\\0.55713&-0.52432&-0.64396\\0.61339&-0.2629&0.74474\end{bmatrix}=1\\&\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.55978&-0.80992&0.17514\\0.55713&-0.52432&-0.64396\\0.61339&-0.2629&0.74474\end{bmatrix}=1\\&\text{The matrix is orthogonal.}\end{align*}$

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Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Determine if the matrix }A=\begin{bmatrix}-1.2228&-0.0071287&-1.0176\\-1.6301&-1.11&-0.23163\\-0.26127&-0.76287&-0.30747\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{Determine if the matrix }A=\begin{bmatrix}-1.2228&-0.0071287&-1.0176\\-1.6301&-1.11&-0.23163\\-0.26127&-0.76287&-0.30747\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-1.2228&-0.0071287&-1.0176\\-1.6301&-1.11&-0.23163\\-0.26127&-0.76287&-0.30747\end{bmatrix}=-1.1685\\&\text{The matrix is not orthogonal.}\end{align*}$ $\displaystyle \begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-1.2228&-0.0071287&-1.0176\\-1.6301&-1.11&-0.23163\\-0.26127&-0.76287&-0.30747\end{bmatrix}=-1.1685\\&\text{The matrix is not orthogonal.}\end{align*}$

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Linear Algebra : Orthogonal Matrices

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