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Find the values of `x , y , z`if the matrix `A=[0 2y z x y-z x-y z]`satisfy the equation `A^T A=I_3`. |
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Answer» It is given that `A=[(0,2y,z),(x,y,-z),(x,-y,z)]` `:. A^(T)=[(0,x,x),(2y,y,-y),(z,-z,z)]` Now, `A^(T)A=I` `implies [(0,x,x),(2y,y,-y),(z,-z,z)][(0,2y,z),(x,y,-z),(x,-y,z)]=[(1,0,0),(0,1,0),(0,0,1)]` `[(0+x^(2)+x^(2),0+xy-xy,0-xz+xz),(0+xy-xy,4y^(2)+y^(2)+y^(2),2yz-yz-yz),(0-xz+zx,2yz-yz-yz,z^(2)+z^(2)+z^(2))]` or `[(2x^(2),0,0),(0,6y^(2),0),(0,0,3z^(2))]=[(1,0,0),(0,1,0),(0,0,1)]` On comparing the corresponding elements, we have `2x^(2)=1` or `x= pm 1/sqrt(2)` `6y^(2)=1` or `y= pm 1/sqrt(6)` `3z^(2)=1` or `z = pm 1/sqrt(3)` |
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