Question

A matrix has eigenvalue (-1) and (-2) the corresponding eigenvectors respectively. Then matrix is (₁₁) and (¹2)

Answer 1

Answer:

Eigenvalues and Eigenvectors

Let A be an n� n matrix over a field F . We recall that a scalar l Î F is said to be an eigenvalue (characteristic value, or a latent root) of A, if there exists a nonzero vector x such that Ax = l x, and that such an x is called an eigen-vector (characteristic vector, or a latent vector) of A corresponding to the eigenvalue l and that the pair (l , x) is called an eigen-pair of A. If l is an eigenvalue of A, the equation: (l I-A)x = 0, has a non-trivial (non-zero) solution and conversely. Thus, this being a homogeneous equation, it follows that l is an eigenvalue of A iff |l I-A| = 0. The expression

fA(x) = |xI-A|