If A is similar to λI for some scalar λ, then A = λI. (T/F)
The statement “If A is similar to λI for some scalar λ, then A = λI” is False
The statement “If A is similar to λI for some scalar λ, then A = λI” is False.
To understand why, it’s important to clarify the concept of similarity between matrices. Two square matrices, A and B, are considered similar if there exists an invertible matrix P such that P^(-1)AP = B. This means that A and B have the same eigenvectors but potentially different eigenvalues.
In this case, if A is similar to λI, it means that there exists an invertible matrix P such that P^(-1)A = λI P. By multiplying both sides of this equation by P^(-1), we can rewrite it as A = λP^(-1)P = λI.
Therefore, if A is similar to λI for some scalar λ, then A is indeed equal to λI. Hence, the statement is True.
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