Linear Algebra Test 2 Review T/F
A matrix A is diagonalizable if A has n eigenvectors
False
If Ax=lambdax for some scalar lambda, then x is an eigenvector of A
False
If Ax=λx for some vector x, then λ is an eigenvalue of A.
False
The eigenvalues of a matrix are on its main diagonal
False
The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (minus−1)r, where r is the number of row interchanges made during row reduction from A to U
False. Reduction to an echelon form may also include scaling a row by a nonzero constant, which can change the value of the determinant. Your answer is correct
The determinant of A is the product of the diagonal entries in A
False. This is only true if A is trangular
If A is invertible, then the columns of A−1 are linearly independent. Explain why.
It is a known theorem that if A is invertible then A−1 must also be invertible. According to the Invertible Matrix Theorem, if a matrix is invertible its columns form a linearly independent set. Therefore, the columns of A−1 are linearly independent.
If C is 6×6 and the equation Cx=v is consistent for every v in set of real numbers ℝ6, is it possible that for some v, the equation Cx=v has more than one solution? Why or why not?
It is not possible. Since Cx=v is consistent for every v in set of real numbers ℝ6, according to the Invertible Matrix Theorem that makes the 6×6 matrix invertible. Since it is invertible, Cx=v has a unique solution.
A row replacement operation does not affect the determinant of a matrix
True. If a multiple of one row of a matrix A is added to another to produce a matrix B, then det B equals det A
If the columns of A are linearly dependent, then det A=0
True. If the columns of A are linearly dependent, then A is not invertible
Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Choose the correct answer below.
Right-multiplication (that is, multiplication on the right) by the diagonal matrix D multiplies each column of A by the corresponding diagonal entry of D. Left-multiplication by D multiplies each row of A by the corresponding diagonal entry of D.
If A can be row reduced to the identity matrix, then A must be invertible.
True; since A can be row reduced to the identity matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is invertible, A is invertible
A product of invertible n×n matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
False; if A and B are invertible matrices then (AB)^-1 = B^-1 A^-1
If A is invertible, then elementary row operations that reduce A to the identity In also reduce A−1 to In
False; if A is invertible, then the row operations required to reduce A to the identity correspond to some product of elementary matrices. Then the row operations required to reduce A^-1 to the identity would correspond to the product
A number c is an eigenvalue of A if and only if the equation (A−cI)x=0 has a nontrivial solution
True
Find a 3×3 matrix B, not the identity matrix or zero matrix, such that ABequals=BA. Choose the correct answer below.
There are infinitely many solutions. Any multiple of I3 will satisfy the expression.
A subset H of set of real numbers ℝn is a subspace if the zero vector is in H.
This statement is false. For each u and v in H and each scalar c, the sum u+v and the vector cu must also be in H
Determine whether the statement "The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A" is true or false
True
Finding an eigenvector of A may be difficult, but checking whether a given vector u is in fact an eigenvector is easy
True
If AP=PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A
True
If A=[a b/ c d] and ad=bc, then A is not invertible
True; if ad=bc then ad-bc=0 and 1/(ad-bc) [d -b/-c a] is undefined
If A is invertible, then the inverse of A^-1 is A itself.
True; since A^-1 is the inverse of A, A^-1A=I=AA^-1. Since A^-1A=I=AA^-1, A is the inverse of A^-1
If ℝn has a basis of eigenvectors of A, then A is diagonalizable
True
How many rows does B have if BC is a 9x8 matrix?
9 rows
If a matrix A is 8x9 and the product AB is 8x4, what is the size of B?
9x4
Solve the equation AB=BC for A, assuming that A, B, and C are square matrices and B is invertible.
A=BCB^-1
A is diagonalizable if A=PDP−1 for some matrix D and some invertible matrix P
False
A is diagonalizable if and only if A has n eigenvalues, counting multiplicities
False
Determine whether the statement "det A^T = (-1)detA" is true or false
False
Determine whether the statement "If A is 3x3 with columns a1,a2,a3 then detA equals the volume of the parallelepiped determined by a1,a2,a3 is T/F
False
Determine whether the statement "A row replacement operation on A does not change the eigenvalues" is true or false
False
If A is diagonalizable, then A has n distinct eigenvalues
False
If A is diagonalizable, then A is invertible
False
If A is invertible, then A is diagonalizable
False
If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues
False
To find the eigenvalues of A, reduce A to echelon form
False. An echelon form of a matrix A usually does not display the eigenvalues of A
If three row interchanges are made in succession, then the new determinant equals the old determinant
False. If three row interchanges are made in succession, then the new determinant equals the negative of the old determinant
If det A is zero, then two rows or two columns are the same, or a row or a column is zero
False; If A = [ 2 6/ 1 3], then det A =0 and the rows and columns are all distinct and not full of zeros
det(A+B)=det A+det B
False; If A= [1 0/ 0 1] and B= [-1 0/ 0 -1], then det(A+B)=0 and det A + det B=2
Det A^-1 = (-1)detA
False; det A ^-1 = (detA) ^-1
Is Nul A =R3
No, because the null space of a 5x8 matrix is a subspace of ℝ8. Although dim Nul A=3, it is not strictly equal to ℝ3 because each vector in Nul A has eighteight components. Each vector in ℝ3 has threethree components. Therefore, Nul A is isomorphic to ℝ3, but not equal.
If the given equation Gx=y has more than one solution for some y in set of real numbersℝn, can the columns of G span set of real numbers ℝn? Why or why not? Assume G is ns×n.
The columns of G cannot span set of real numbers ℝn. According to the Invertible Matrix Theorem, if Gx=y has more than one solution for some y in set of real numbers ℝn, that makes the matrix G non invertible.
A subspace of set of real numbers ℝn is any set H such that (i) the zero vector is in H, (ii) Bold uu, Bold vv, and Bold uu+Bold vv are in H, and (iii) c is a scalar and cBold uu is in H.
The statement is false. Conditions (ii) and (iii) must be satisfied for each Bold uu and Bold vv in H, which is not specified in the given statement.
This statement is true. For an m×n matrix A, the solutions of Ax=0 are vectors in ℝn and satisfy the properties of a vector space
This statement is false. The column space of A is the set of all b for which Ax=b has a solution
B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A
This statement is false. The columns of an echelon form of a matrix are often not in the column space of the original matrix
Given vectors v1....vp in ℝn, the set of all linear combinations of these vectors is a subspace of ℝn.
This statement is true. This set satisfies all properties of a subspace.
A matrix A is not invertible if and only if 0 is an eigenvalue of A
True
An eigenspace of A is a null space of a certain matrix
True
A steady-state vector for a stochastic matrix is actually an eigenvector
True
If A,B and C are nxn invertible matrices, does the equation C^-1(A+X)B^-1= In have a solution X
X=CB-A
Is Col A = R5
Yes, because the column space of a 5x8 matrix is a subspace of ℝ5. There is a pivot in each row, so the column space is 5-dimensional. Since any 5-dimensional subspace of ℝ5 is ℝ5, Col A=ℝ5
