True/False Test 3
(AB)−1=A−1B−1.
False
A 5x5 real matrix has even number of real eigenvalues
False
A is diagonalizable if and only if A has n real eigenvalues counting multiplicity
False
If A is diagonalizable, then A is invertible
False
If det A is zero, then two columns of A must be the same, or all of the elements in a row or column of A are zero.
False
If the columns of A are linearly independent, then detA= 0
False
Row operations on a matrix do not change its eigenvalues
False
Suppose A is a 3×3 matrix and λ is a real number with the property that the equation Ax=λx is satisfied by some nonzero vector x. For each question, select true or false. If a statement does not even make sense, select false as your answer. A is not invertible
False
Suppose A is a 3×3 matrix and λλ is a real number with the property that the equation Ax=λx is satisfied by some nonzero vector x. For each question, select true or false. If a statement does not even make sense, select false as your answer. A−λ is invertible.
False
The characteristic polynomial of the zero matrix is 0
False
The determinant of a triangular matrix is the sum of the entries of the main diagonal
False
The eigenvalues of A are the entries on its main diagonal
False
λ is an eigenvalue of a matrix A if A−λI has linearly independent columns.
False
A square matrix with two identical columns can be invertible.
No
if the linear transformation T(x) = Ax is 1 to 1, then the columns of A forms a linearly dependent set
No
A determinant of an n×n matrix can be defined as a sum of multiples of determinants of (n−1)×(n−1) submatrices.
True
A number c is an eigenvalue of A if and only if (A−cI)v=0 has a nontrivial solution.
True
A real eigenvalue of a real matrix always has at least one corresponding real eigenvector
True
A row replacement operation does not affect the determinant of the matrix
True
If A is nxn and A has n distinct eigenvalues then the corresponding eigenvectors of A are linearly independent
True
If AA is a 4×4 matrix with characteristic polynomial λ4+λ3+λ2+λ, then A is not invertible.
True
If the characteristic of a 2x2 matrix is λ2−5λ+6, then the determinant is 6.
True
If two columns of A are the same, then the determinant of that matrix is zero
True
If v is an eigenvector of A, then cv is also an eigenvector of A for any number c≠0
True
Suppose A is a 3×3 matrix and λ is a real number with the property that the equation Ax=λx is satisfied by some nonzero vector x. For each question, select true or false. If a statement does not even make sense, select false as your answer. A- XI is not invertible
True
The (i, j) minor of a matrix A is the matrix Aij, obtained by deleting row i and column j from A
True
The cofactor expansion of det A along the first row of A is equal to the cofactor expansion of det A
True
if there is a basis of Rn consisting of eigenvectors of A, then A is diagonalizable
True
If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivots.
Yes
If −A is not invertible, then A is also not invertible.
Yes
The product of any two invertible matrices is invertible.
Yes
(A+B)2= A2 + B2 + 2AB
false
A+ B invertible
false
There exists a real 2x2 matrix with the eigenvalues of i and 2i
false
det(A +B) = det(A) + det(b)
false
if an nxn matrix A has fewer than n distinct real eigenvalues then A is diagonalizable
false
If the equation Ax=0 has the trivial solution, then the columns of A span Rn
maybe
(In−A)(In+A)=In−A2
true
A is invertible if and only if 0 is not an eigenvalue of A
true
Is A7 invertible
true
The absolute value of the determinant of A equals the volume of the parallelepiped determined by the columns of A
true
every real 3x3 matrix must have at least one real eigenvalue
true
If A is invertible, then the equation Ax=b has exactly one solution for all b in Rn
yes
If A2 is row equivalent to the n×n identity matrix, then the columns of A span Rn
yes
