Determine if the statement is true or false, and justify your answer. A 2 times
ID: 3031750 • Letter: D
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.