Final answer:
The correct answer to whether a real matrix can have eigenvalues a and b is 'A. There exists a real matrix with the eigenvalues a and b.' This is because diagonal matrices have their diagonal entries as eigenvalues and a 2x2 diagonal matrix with a and b as its diagonal entries will have these as its eigenvalues.
Step-by-step explanation:
Understanding the properties of eigenvalues and eigenvectors in linear algebra is crucial when dealing with matrices, particularly real matrices.
When addressing the question of how many real eigenvalues a real matrix has, it's important to recall the fundamental theorem of algebra which states that every polynomial equation of degree n has n roots in the complex number system, where some of the roots may be real or non-real complex numbers.
Interestingly, when dealing with real matrices, complex eigenvalues always come in conjugate pairs. This knowledge helps us to deduce the fact that real matrices can indeed have an even number of real eigenvalues.
Now let's examine the given statements:
- A. There exists a real matrix with the eigenvalues a and b - This is true, as any 2 x 2 matrix with a and b as its diagonal entries and zeroes elsewhere will have these as its eigenvalues.
- B. A real eigenvalue of a real matrix always has at least one corresponding real eigenvector - This is also true, as real eigenvalues of real matrices do correspond to real eigenvectors.
- C. Every real matrix must have at least one real eigenvalue - This is false, as real matrices can have complex eigenvalues, particularly in the case of odd-dimensional matrices with complex conjugate pairs of eigenvalues.
With the understanding of these concepts, we can address the multiple choice question:
Answer: The correct option is A. There exists a real matrix with the eigenvalues a and b. This is based on the property that diagonal entries of a diagonal matrix are indeed its eigenvalues.