The statements below belong to two groups.In each group, all of the statements are equivalent to one another.Write the statements in the correct group.In all of the statements, A is a square n × n matrix.
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A is nonsingular.
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A is singular.
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The columns of A are linearly dependent.
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The columns of A are linearly independent.
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The columns of A span R^n.
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The subspace spanned by the columns of A is not
all of R^n.
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Ax=0 has only the trivial solution
x=0.
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Ax=0 has an infinite number of solutions
x.
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There is a b in R^n such that Ax=b
has no solution x.
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For all b in R^n, Ax=b has a solution
x.
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A is row equivalent to a matrix with a row of zeros.
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A is row equivalent to a nonsingular matrix.
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N(A) = {0}.
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There are nonzero vectors in N(A).
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A has no inverse.
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A has an inverse matrix
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The dimension of N(A) is greater than 0.
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The dimension of the null space of A is 0.
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The dimension of the column space of A is n.
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The dimension of the column space of A is less than
n.
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The dimension of the row space of A is n.
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The dimension of the row space of A is less than
n.
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Group 1
Group 2
det(A)=0
det(A) is not = 0