Linear Algebra Test 2
T/F If A is invertible, then the inverse of A^-1 is A itself
True
T/F If A= [a b c d] and ad=bc, then A is not invertible
True
T/F A is diagonizable if and only if A has n eigenvalues, counting multiplicities
false
T/F A subset H of Rn is a subspace if the zero vector is in H
false
T/F A subspace of R^n is any set H such that (i) the zero vector is in H, (ii) u, v, and u+v are in H (iii) c is a scalar and cu is in H
false
T/F An elementary row operation on A does not change the determinant
false
T/F If A is 3x3, with olumns a1, a2, a3, then detA equals the volume of the parallelepiped determined by a1, a2, a3
false
T/F The eigenvalues of a matrix are on its main diagonal
false
T/F a row replacement operation on A does not change the eigenvalues
false
T/F det(A+B) = detA + detB
false
T/F The columns of an invertible nxn matrix form a basis for Rn
true
T/F A is diagonalizable if A=PDP^-1 for some matrix D and some invertible matrix P
false
T/F If A is diagonizable, then A has n distinct eigenvalues
false
T/F If A is invertible, then A is diagonizable
false
T/F If A is invertible, then elementary row operations that reduce A to the identity In also reduce A^-1 to In
false
T/F If Ax=lambdax for some scalar lambda, then x is an eigenvector of A
false
T/F If Ax=lambdax for some vector x, then lambda is an eignenvalue of A.
false
T/F If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A
false
T/F If C is 6x6 and the equation Cx=v is consistent for every v in R^6, is it possible that for some v, the equation Cx=v has more than one solution?
false
T/F If detA is zero, then two rows or two columns are the same, or a row or a column is zero
false
T/F If lambda + 5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A
false
T/F If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues
false
T/F The cofactor expansion of detA down a column is the negative of the cofactor expansion along a row
false
T/F The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^R, where r is the number of row interchanges made during row reduction from A to U
false
T/F The determinant of a triangular matrix is the sum of the entries of the main diagonal.
false
T/F if three row interchanges are made in succession, then the new determinant equals the old determinant
false
T/F is it possible for a 5x5 matrix to be invertible when its columns do not span R^5?
false
T/F the column space of a matrix A is the set of solutions of Ax=b
false
T/F the determinant of A is the product of the diagonal entries
false
T/F the set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of Rm
false
det(A^-1) = (-1)detA
false
det(A^T) = (-1)detA
false
When is a square upper triangular matrix invertible?
A square upper triangular matrix is invertible when all entries on its main diagonal are nonzero. If all of the entries on its main diagonal are nonzero, then the nxn matrix has n pivot positions
T/F Can a square matrix with two identical columns be invertible?
False
Explain why the columns of an nxn matrix A span R^n when A is invertible
Since A is invertible, for each b in R^n the equation Ax=b has a unique solution. Since the equation Ax=b has a solution for all b in R^n, the columns of A span R^n
If the Nul is not in the zero subspace of a matrix, then the matrix
has nontrivial solution
A is a 3x3 matrix with two eigenvalues. Each space is one-dimensional. Is A diagonizable?
no
If the given equation Gx=y has more than one solution for some y in R^n, can columns of G span R^n
no
If a column space does not equal the number of rows, then the Nul contains a
nonzero vector
(detA)(detB) = det(AB)
true
If v1 and v2 are in Rn, the S=span{v1, ..., v2} is the same column space of the matrix A=[v1 . . . . vp]
true
T/F A matrix A is not invertible if and only if 0 is an eignenvalue of A
true
T/F A number c is an eigenvalue of A if and only if the equation (A-cI)x=0 has a nontrivial solution
true
T/F A row replacement operation does not affect the determinant of a matrix
true
T/F An eigenspace of A is a null space of a certain matrix
true
T/F Finding an eigenvector of A may be difficult, but checking whether a given vector u is in fact an eigenvector is easy
true
T/F Given vectors v1, ..., vp, in Rn, the set of all linear combinations of thee vectors is a subspace of Rn
true
T/F If A can be row reduced to the identity matrix, then A must be invertible
true
T/F If A is invertible then A^-1 must also be invertible
true
T/F If A is invertible, then the columns of A^-1 are linearly independent
true
T/F If AP=PD, with D diagonal, then the nonzero columns of P must be eigenvectors
true
T/F If R^n has a basis of eigenvectors of A then A is diagonalizable
true
T/F If the columns of A are linearly dependent, then det(a)=0
true
T/F Row operations do not affect linear dependence relations among the columns of a matrix
true
T/F The multiplicity of a root r od the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A
true
T/F The null space of an mxn matrix is a subspace of Rn
true
What is the rank of a 7x9 matrix whose null space is five dimensional?
4
If a matrix is 4x8 and the product AB is 4x4, what is the size of B?
8x4
How many rows does B have if BC is a 9x3 matrix?
9
How many rows does B have if BC is a 9x5 matrix?
9
If a matrix A is 8x4 and the product Ab is 8x5, what is the size of B
4x5
What is the rank of a 6x8 matrix whose null space is three dimensional?
5
If a matrix A is 4x5 and the product AB is 4x8, what is the size of B
5x8
How many rows does B have if BC is a 7x9 matrix
7
If the subspace of all solutions of Ax=0 has a basis consisting of six vectors and if A is a 7x9 matrix, what is the rank of A
3
If a matrix A is 3x3 and the product AB is 3x5, what is the size of B?
3x5
If the rank of a 7x9 matrix A is 6, what is the dimension of the solution space Ax=0?
3
(T/F) The determinant of A is the product of the diagonal entries in A
false
A matrix A is diagonizable if A has n eigenvectors
false
If A is diagonizable, then A is invertible
false
