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Matrix Operations and Properties
Matrix Operations and Properties
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If a 3x3 matrix has eigenvalues 5, 2, and 0, what can you conclude about the matrix?
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A.
The matrix must be diagonal because having three distinct eigenvalues forces the eigenvectors to form a standard basis for the space
B.
The matrix is singular (determinant = 0), has rank 2, and trace 7, because one zero eigenvalue makes it non-invertible with a nontrivial null space
C.
Nothing definitive can be concluded without knowing the eigenvectors, since eigenvalues alone do not determine any property of the matrix
D.
The matrix is invertible with determinant 10, since the product of the nonzero eigenvalues determines whether a matrix can be inverted
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