Midterm2practice_sol - Math 410 Midterm 2 practice solutions Problem 1 Given a matrix A if it is complete(i.e if it has a full set of eigenvectors one

# Midterm2practice_sol - Math 410 Midterm 2 practice...

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Math 410 — Midterm 2 practice solutions Problem 1: Given a matrix A , if it is complete (i.e. if it has a full set of eigenvectors), one can put the eigenvalues of A in a diagonal matrix D and the corresponding eigenvectors as columns in a matrix V and have A = V DV - 1 This is called diagonalizing A . So we just have to write D = 2 0 0 3 ; V = 1 2 1 - 1 and calculate A = V DV - 1 = 1 3 8 - 2 - 1 7 This should be the matrix we needed. You can verify this by calculating the eigenvalues and eigenvectors of A . Problem 2: To find the eigenvalues we need to solve the characteristic equation det ( A - λ I ) = det 2 - λ 1 1 0 4 - λ 0 6 1 1 - λ = 0 where A is the given matrix. To calculate the determinant of this 3 × 3 matrix we can use the cofactor expansion, on the second row, to get (4 - λ )[(2 - λ )(1 - λ ) - 6] = 0 (4 - λ )( λ + 1)( λ - 4) = 0 So the eigenvalues are λ = 4 (double root) and λ = - 1. For λ = 4 we look for the eigenvectors by finding the nullspace of ( A - 4 I ) which gives us the single 1
eigenvector 1 0 2 (so the matrix is not complete.) For
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