The statement belowSolutionthe statement is true I remember

Solution

the statement is true.

I remember solving the same question once before.

consider an example where the characteristic polynomial of (sI-A) is given by

s³ + 3s² + 5s + 7

to obtain eigenvalues equate the polynomial to 0

you will get s = -2.18,-0.41,-0.41

so when s assumes any of these values, the matrix (sI-A) becomes singular and hence- non-invertible.

so, If (sI-A) is singular, then s is an eigenvalue of A.

This also means that (sI-A) is invertible if (s) is not an eigenvalue of A.

also

in above example, if s>-0.41, then the matrix (sI-A) will always be invertible

hence

the statement

(sI-A) will be invertible if (s) is larger than the largest eigenvalue of A. is also correct

so overall the given statement is always true