Suppose that A is a 7 x 7 matrix whose only distinct eigenvalues are 1, 2, and 3. Let cA be the characteristic polynomial of A. The degree of CA(x) is The fewest basic 1-eigenvectors that A can have is and the most basic 1-eigenvectors that A can have is Now, suppose in addition that A is diagonalizable and that A has three basic 1-eigenvectors. Under these assumptions, fill in the blanks in the exponents of CA(x) below, and determine the five ranks listed below. 1)0(x CA(x) = (x-1) (x - 2)²(x-3) rank(I7- A) = rank (217 - A) = rank (317- A) = | rank (417 - A) = rank(A) =

Q: Show that an n x n matrix A is invertible if and only if 0 is not an eigenvalue of A.

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