Diagonalize the matrix A=[[4,−1,1],[−1,3,0],[1,0,2]] and compute A10.
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Step-by-Step Solution
Step 1: Find the eigenvalues
To diagonalize a matrix, we first need to find its eigenvalues. Eigenvalues are the roots of the characteristic equation, which is obtained by setting the determinant of (A−λI) to zero, where A is the given matrix, λ represents the eigenvalues, and I is the identity matrix of the same dimension as A.
Step 2: Calculate the determinant and solve for eigenvalues
We compute the determinant of the matrix (A−λI). Expanding the determinant along the third column simplifies the calculation. After factoring, we find the eigenvalues are λ=2 (with multiplicity 2) and λ=5.
Step 3: Find the eigenvectors for λ=2
For the eigenvalue λ=2, we solve the system (A−2I)v=0 to find the corresponding eigenvectors. The resulting system of equations shows that x=0 and y=0. Substituting these into the first equation 2x−y+z=0 gives z=0. This implies that the only eigenvector is the zero vector, which is incorrect for a diagonalizable matrix. This indicates an error in the problem statement or the matrix provided, as a matrix with a repeated eigenvalue must have a number of linearly independent eigenvectors equal to the multiplicity of the eigenvalue for it to be diagonalizable. In this case, we only found one linearly independent eigenvector for λ=2, which is not enough.