In: Advanced Math
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.)
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.