Suppose that A is an n × n matrix satisfying A3 - 3A + 2A-3l =...

Suppose that A is an n × n matrix satisfying A3 - 3A + 2A-3l = 0.

Show that A is invertible by the definition of invertible. (Hint: Review the definition of invertible, and then describe the inverse in terms of the matrix A - you don’t need to know what A is to answer this question.)

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Expert Solution

Solution:

Definition used:

A square matrix os order is invertible if and only if is NOT and eigenvalue of .

Given that the matrix satisfies the equation

Hence, by Cayley-Hamilton theorem, we have that the characteristic equation of the matrix is

which clearly shows that is not a root of this characteristic equation due to the presence of the constant term which in turn mean that . Consequently, is NOT an eigenvalue of and hence is invertible which can be obtained as follows:

Hence the required result. If you still have any doubt please post your comment Kindly rate the answer accordingly.

Colby Messinger answered 1 year ago

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