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If `A = [(0,-tan(alpha//2)),(tan(alpha//2),0)]` and `I` is a `2xx2` unit matrix, prove that `I+A=(I-A)[(cosalpha,-sinalpha),(sinalpha,cosalpha)]` |
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Answer» Since, `I=[(1,0),(0,1)]` and given `A = [(0,-tan(alpha//2)),(tan(alpha//2),0)]` `therefore" " 1+A = [(1,-tan(alpha//2)),(tan(alpha//2),1)]` `RHS = (I-A)[(cosalpha,-sinalpha),(sinalpha,cosalpha)]` `=[(1,tan(alpha//2)),(-tan(alpha//2),1)][(cosalpha,-sinalpha),(sinalpha,cosalpha)]` `=[(1,tan(alpha//2)),(-tan(alpha//2),1)]` `[((1-tan^(2)alpha//2)/(1tan^(2)(alpha//2)),(-2tan(alpha//2))/(1+tan^(2)(alpha//2))),((2tan(alpha//2))/(1+tan^(2)(alpha//2)),(1-tan^(2)(alpha//2))/(1+tan^(2)(alpha//2)))]` `RHS = [(1,lambda),(-lambda,1)][((1-lambda^(2))/(1+lambda^(2)),(-2lambda)/(1+lambda^(2))),((2lambda)/(1+lambda^(2)),(1-lambda^(2))/(1+lambda^(2)))]` `[((1-lambda^(2)+2lambda^(2))/(1+lambda^(2))(-2lambda+lambda(1-lambda^(2)))/(1+lambda^(2))),((-lambda(1-lambda)^(2)+2lambda)/(l+lambda^(2))(2lambda^(2)+1-lambda^(2))/(1+lambda^(2)))]` `[((1+lambda^(2))/(1+lambda^(2))( lambda(1+lambda)^(2))/(1+lambda^(2))),((lambda(1+lambda^(2)))/(1+lambda^(2))(1+lambda^(2))/(1+lambda^(2)))]=[(1,-lambda),(lambda,1)]` `=[(1,-tan(alpha//2)),(-tan(alpha//2),1)][therefore lambda =(alpha//2)]` `=I+A` `=LHS` |
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