Which of the following is an orthogonal matri | vedclass

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(A) square matrix $A$ is orthogonal if $AA^T = A^TA = I$,where $I$ is the identity matrix. This implies that the sum of squares of elements in each ro

Vedclass · Accepted Answer

$\begin{bmatrix} 6/7 & 2/7 & -3/7 \ 2/7 & 3/7 & 6/7 \ 3/7 & -6/7 & 2/7 \end{bmatrix}$

Vedclass · Answer

(A) square matrix $A$ is orthogonal if $AA^T = A^TA = I$,where $I$ is the identity matrix. This implies that the sum of squares of elements in each row (or column) is $1$,and the sum of products of corresponding elements of any two distinct rows (or columns) is $0$.Let us check option $A$:Row $1$: $(6/7)^2 + (2/7)^2 + (-3/7)^2 = (36+4+9)/49 = 49/49 = 1$.Row $2$: $(2/7)^2 + (3/7)^2 + (6/7)^2 = (4+9+36)/49 = 49/49 = 1$.Row $3$: $(3/7)^2 + (-6/7)^2 + (2/7)^2 = (9+36+4)/49 = 49/49 = 1$.Dot product of Row $1$ and Row $2$: $(6/7)(2/7) + (2/7)(3/7) + (-3/7)(6/7) = (12+6-18)/49 = 0/49 = 0$.Dot product of Row $2$ and Row $3$: $(2/7)(3/7) + (3/7)(-6/7) + (6/7)(2/7) = (6-18+12)/49 = 0/49 = 0$.Dot product of Row $1$ and Row $3$: $(6/7)(3/7) + (2/7)(-6/7) + (-3/7)(2/7) = (18-12-6)/49 = 0/49 = 0$.Since all conditions are satisfied,the matrix in option $A$ is orthogonal.

Which of the following is an orthogonal matrix?

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