Question

It can be shown that the eigenvalues of the matrix A=[a 0 / 0 b] are λ₁ =a and λ₂=b. Assume a≠b, so the eigenvalues of the matrix are distinct. Now, continue with the question or discussion related to the properties or implications of these distinct eigenvalues.

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Answer 1

The distinct eigenvalues of a matrix have several implications, including the matrix being invertible and diagonalizable.

Step-by-step explanation:

The question pertains to the properties or implications of distinct eigenvalues of a matrix. When the matrix A has distinct eigenvalues, it means that the matrix can be diagonalized, which has several implications. One implication is that the matrix is invertible, meaning it has an inverse matrix. Another implication is that the matrix can be expressed as a diagonal matrix, where the eigenvalues are the diagonal entries.

For example, if A=[a 0 / 0 b] and a≠b, then the eigenvalues are λ₁ =a and λ₂=b. This means that A can be diagonalized as D=[λ₁ 0 / 0 λ₂] where D is a diagonal matrix.