Math 369 T/F
Assume that V s the "vector" space formed by all real square matrices of order n with the usual matrix addition and scalar multiplication. Assume that W is the subset of V that consists of all invertible matrices. Then W is a subspace of V
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For a 3x3 orthogonal matrix Q, all its eigenvalues are real numbers
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For a 3x3 real matrix A, if the determinant is zero, then 0 must be an eigenvalue
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For a 5x6 matrix A, if rank(A)=2, then nullity(A^T)=4
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For a 5x6 matrix A, if rank(A)=2, then there are 2 parameters in the general solution of Ax=0
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For a 5x7 matrix A, if rank(A)=3, then the dimension of the null space of A is also 3
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For a 5x7 matrix A, if rank(A)=3, then there are 3 parameters in the general solution of Ax=0
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For a 5x7 matrix, if rank(A)=3, then there are 3 parameters in the general solution of Ax=0
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For a square matrix A, if det(A)=3, then det(A^-1)=-3
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For a square matrix, if the determinant is zero, then its column vectors are linearly independent
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For any matrix A, if 1 is an eigenvalue, then I-A is an invertible matrix
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For any matrix A, if λ is an eigenvalue, then λ I-A is an invertible matrix
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For any matrix A, if λ is an eigenvalue, then λI-A is an invertible matrix
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For order 3 square matrices A,B, we always have (A+B)^2 = A^2 +2AB + B^2
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For order 7 square matrices A,B, we always have (A+B)^2=A^2+2AB+B^2
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If A is a 5x7 matrix, then rank (A^T) + nullity (A^T) = 7
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If A is a square matrix of order 3 and det(A)=6, then det(-2A)=-12
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If A is a square matrix of order 5 and det(A)=3, then det(A^-1)=-3
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If A is an 8x8 matrix and A^2=0, then A must be a zero matrix
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If A is an order 2018 square matrix and A^2=0, then A must be a zero matrix
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If A,B are both invertible square matrices of the same order, then (AB)^-1=A^-1B^-1
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If Q is an order 3 orthogonal matrix, then we must have det(Q)=-1
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If the algebraic multiplicity of an eigenvalue is 2, then we can find two linearly independent eigenvectors in its eigenspace
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Multiplying a row of an augmented matrix through by zero is an acceptable elementary row operation
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The set of upper triangular nxn matrices is a subspace of the vector space of all nxn matrices
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There exists a real 5x5 matrix with no real eigenvalues
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These three vectors v1=(1,0,0), v2=(1,1,0), v3=(1,1,1) are linearly dependent
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A single linear equation with two or more unknowns has infinitely many solutions
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A square matrix A is invertible if and only if det(A) is not equal to 0
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All leading 1's in a matrix in row echelon form must occur in different columns
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All leading 1's in reduced row echelon form must occur in different columns
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An order 2019 real square matrix has at least one real eigenvalue
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For a 3x3 real matrix A, it has at least one real eigenvalue
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For a 3x3 real matrixA, if 2+i is an eigenvalue, then 2-i is also an eigenvalue
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For a 3x3 real symmetric matrix A, all its eigenvalues are real numbers
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For a 5x6 matrix A, if rank(A)=2, then nullity(A)=4
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For a 5x6 matrix A, if rank(A)=2, then the dimension of its column space is 2
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For a 5x6 matrix A, if rank(A)=2, then there are 4 parameters in the general solution of Ax=0
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For a 5x7 matrix A, if rank(A)=3, then the dimension of te row space of A is 3
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For a 5x7 matrix A, if rank(A)=3, then there are 4 parameters in the general solution of Ax=0
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For a 5x7 matrix, if rank(A)=3, then nullity(A)=4
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For an order 3 square matrix A, if 0 is one eigenvalue of A, then det(A)=0
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For an order n square matrix A, if rank(A)<(=)n, then its column vectors are linearly dependent
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For any two vectors u, v in the same inner product space, we always have |<u,v>|<(=)||u||||v||
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If A is a square matrix of order 3 and det(A)=6, then A is invertible
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If A is a square matrix of order 5 and det(A)=3, then det(-A)=-3
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If A is an order 3 upper triangular matrix and all the entries on the main diagonal are nonzero, then A is invertible
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If A,B are nxn matrices and AB=I (order n identity matrix), then AB=BA
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If A,B are nxn matrices and AB=I (order n identity matrix), then BA=I
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If a homogeneous linear system has n equations and n+1 unknowns, then it surely has nontrivial solutions
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If a set of five vectors contains the zero vector, then these five vectors are linearly dependent
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If a square matrix A is invertible, then the homogeneous linear system Ax=0 has only the trivial solution
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If matrix A is invertible, then A^T is also invertible
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If matrix A is invertible, then det(A) is not equal to 0
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If u,v are two vectors in R^n, then |u.v|<(=)||u||||v||
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If u,v,w are vectors in R^n, then ||u+v+w||<(=)||u||+||v||+||w||
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Let A be an order 2018 real matrix. If 10+26i is a complex eigenvalue of A, then 10-26i is also an eigenvalue
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Three vectors v1=(1,0,0), v2=(1,1,0), v3=(1,1,1) form a basis or R^3
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Two vectors are linearly dependent if and only if one vector is a multiple of the other
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