Linear Algebra Test 2
To find the eigenvalues of A, reduce A to echelon form. Choose the correct answer below.
F it must be rref
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. Reduction to an echelon form may also include scaling a row by a nonzero constant, which can change the value of the determinant.
If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues. Choose the correct answer below.
False. There may be linearly independent eigenvectors that both correspond to the same eigenvalue.
The determinant of A is the product of the diagonal entries in A.
False. This is only true if A is trangular.
Is it possible for a 5×5 matrix to be invertible when its columns do not span set of real numbers ℝ5? Why or why not?
It is not possible; according to the Invertible Matrix Theorem an n×n matrix cannot be invertible when its columns do not span set of real numbers ℝn.
Can a square matrix with two identical columns be invertible? Why or why not?
The matrix is not invertible. If a matrix has two identical columns then its columns are linearly dependent. According to the Invertible Matrix Theorem this makes the matrix not invertible.
If A is diagonalizable, then A has n distinct eigenvalues.
The statement is false. A diagonalizable matrix can have fewer than n eigenvalues and still have n linearly independent eigenvectors.
The dimension of the column space of A is rank A. Choose the correct answer below.
True
The dimensions of Col A and Nul A add up to the total number of columns in A. Choose the correct answer below.
True
The null space of an m×n matrix is a subspace of set of real numbers ℝn.
True
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.
What is the subspace of all solutions that has a basis consisting of five vectors and if A is a 5x7 matrix, what is the rank of A?
2
What is the rank of a 6x9 matrix whose null space is five dimensional
4
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
A row replacement operation does not affect the determinant of a matrix
True
when a row is replaced by itself plus k times another row how does it affect the determinant
it does not affect the determinant
When rows are swapped how does it change the determinant
it swaps the sign of the determinant
If the determinant of the matrix is zero is it invertible?
no
is det(5A) the same as 5det(A)?
no
If the determinant of the matrix is zero is the matrix linearly dependent?
yes
The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row.
False, because the determinant of A can be computed by cofactor expansion across any row or down any column. Since the determinant of A is well defined, both of these cofactor expansions will be equal.
A steady-state vector for a stochastic matrix is actually an eigenvector. Choose the correct answer below.
True
Finding an eigenvector of A may be difficult, but checking whether a given vector u is in fact an eigenvector is easy. Choose the correct answer below.
True
If A is invertible, then the inverse of A^−1 is A itself.
True
The columns of an invertible n×n matrix form a basis for set of real numbers ℝ^n
True. The columns of an invertable matrix are linearly independant
det(A+B)=det A+det B
False.
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 Upper A Superscript negative 1 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 Upper A Superscript negative 1 A−1 are linearly independent.
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.
If the rank of 6X9 matrix A is 3 what is the dimension of the solution space?
6
If det A is zero, then two rows or two columns are the same, or a row or a column is zero.
False
det (A^-1) = (-1)detA
False, Det A^-1 = (detA)^-1
The determinant of a triangular matrix is the sum of the entries on the main diagonal.
False, because the determinant of a triangular matrix is the product of the entries along the main diagonal.
The eigenvalues of a matrix are on its main diagonal. Choose the correct answer below.
False, it has to be a triangular matrix
The dimension of Nul A is the number of variables in the equation Ax equals =0. Choose the correct answer below.
False, it is the number of free variables
If Ax=λx for some scalar lambda λ, then x is an eigenvector of A. Choose the correct answer below.
False, not enough info. The vector must be nonzero
f Ax=λx for some vector x, then λ is an eigenvalue of A. Choose the correct answer below.
False,not enough info. Must have nontrivial solution
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.z
Can a square matrix with two identical columns be invertible? Why or why not?
No
Is it possible for a 5×5 matrix to be invertible when its columns do not span set of real numbers ℝ5? Why or why not?
No
Explain why the columns of an nxn matrix A span Rn when A is invertible
Since A is invertible, for each b in Rn the equation Ax = b has a unique solution. Since the equation Ax = b has a solution for all b in Rn, the columns of A span Rn
If A is diagonalizable, then A is invertible. Choose the correct answer below.
The statement is false because invertibility depends on 0 not being an eigenvalue. A diagonalizable matrix may or may not have 0 as an eigenvalue.
A is diagonalizable if and only if A has n eigenvalues, counting multiplicities. Choose the correct answer below.
The statement is false because the eigenvalues of A may not produce enough eigenvectors to form a basis of set of real numbers R Superscript nℝn.
A matrix A is diagonalizable if A has n eigenvectors.
The statement is false. A diagonalizable matrix must have n linearly independent eigenvectors.
. If set of real numbers R Superscript nℝn has a basis of eigenvectors of A, then A is diagonalizable. Choose the correct answer below.
The statement is true because A is diagonalizable if and only if there are enough eigenvectors to form a basis of set of real numbers R Superscript nℝn.
If APequals=PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A.
The statement is true. Let v be a nonzero column in P and let lambdaλ be the corresponding diagonal element in D. Then APequals=PD implies that Avequals=lambdaλv, which means that v is an eigenvector of A.
A matrix A is not invertible if and only if 0 is an eigenvalue of A. Choose the correct answer below.
True
A number c is an eigenvalue of A if and only if the equation (A−cI)x=0 has a nontrivial solution. Choose the correct answer below.
True
An eigenspace of A is a null space of a certain matrix. Choose the correct answer below.
True
Row operations do not affect linear dependence relations among the columns of a matrix.
True
If {a b, c d} (that is a 2x2 matrix) and ad = bc then A is not invertible
True, it is undefined
If A is invertible, then A is diagonalizable.
he statement is false. An invertible matrix may have fewer than n linearly independent eigenvectors, making it not diagonalizable.
