If 0 is an Eigen value of a matrix A, then the set of columns of A is
Question
If 0 is an Eigen value of a matrix A, then the set of columns of A is
Solution
If 0 is an eigenvalue of a matrix A, then the set of columns of A is linearly dependent.
Here's why:
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An eigenvalue of a matrix A is a scalar (λ) such that when it is subtracted from each entry on the main diagonal of A and then the determinant is calculated, the result is zero. This is represented by the equation det(A - λI) = 0, where I is the identity matrix.
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If 0 is an eigenvalue, then the equation becomes det(A - 0I) = det(A), which means the determinant of A is zero.
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A determinant of zero indicates that the rows (and therefore the columns) of the matrix are linearly dependent.
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Linear dependence of columns means that one column can be represented as a linear combination of the other columns.
So, if 0 is an eigenvalue of a matrix A, then the set of columns of A is linearly dependent.
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