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To solve this system, we will first find the eigenvalues and eigenvectors of the matrix A = [9 0 -8; 4 -3 0; 4 0 -3]. The eigenvalues can be found by solving the characteristic equation:

det(A - lambda*I) = 0

where I is the 3x3 identity matrix. Simplifying this equation, we get:

det([9-lambda 0 -8] [4 -3-lambda 0] [4 0 -3-lambda]) = 0

Expanding the determinant using the first row, we get:

(9-lambda) * det([-3-lambda 0] [0 -3-lambda]) - 8 * det([4 -3-lambda] [4 0]) = (9-lambda) * (lambda^2 + 6lambda + 9) - 8(-12-4*(3+lambda)) = lambda^3 - 9lambda^2 - 2lambda + 36 = 0

This cubic equation can be factored as (lambda-3)^2 * (lambda+4) = 0, so the eigenvalues are lambda1 = 3 (with algebraic multiplicity 2) and lambda2 = -4.

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