Determine if the statement is true or false, and justify your answer. A 2 times

Question

Determine if the statement is true or false, and justify your answer. A 2 times 2 matrix A with real eigenvalues has a rotation-dilation matrix hidden within it. False. A needs to be a complex matrix with complex eigenvalues. True. If lambda_1 and lambda_2 are the eigenvalues of A, then [lambda_1 lambda_2 -lambda_2 lambda_1] is the rotation-dilation matrix hidden within it. False. A needs to be a complex matrix with real eigenvalues. True. If lambda_1 and lambda_2 are the eigenvalues of A, then [lambda_1 -lambda_2 lambda_2 lambda_1] is the rotation-dilation matrix hidden within it. False. A needs to be a real matrix with complex eigenvalues. Determine if the statement is true or false, and justify your answer. The amount of dilation imparted by a rotation-dilation matrix A is equal to |lambda|, where A is an eigenvalue of A. False, since the dilation is Squareroot a^2 + b^2 notequalto a^2 + b^2 = |lambda| where lambda = a plusminus ib is an eigenvalue of A. True, since the dilation is a^2 + b^2 = |lambda| where lambda = a plusminus ib is an eigenvalue of A. True, since the dilation is Squareroot a^2 + b^2 = |lambda| where lambda = a plusminus ib is an eigenvalue of A. False, since the dilation is a^2 + b^2 notequalto Squareroot a^2 + b^2 = |lambda| where lambda = a plusminus ib is an eigenvalue of A.

Explanation / Answer

A rotation dilation is a composition of a rotation by angle arctan(a/b) and a dilation by a factor sqrt (a^2 +b^2).

Now if we know = a +- bi the

|| = sqrt(a^2 +b^2)

which is equal to the above defined in the definition of rotation dialation.

Hence given statement is true.

Third option is the correct one.

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