LA TF 2
How many rows does B have if BC is a 4 times 6 matrix?
4 rows
If a matrix A is 6 times 9 and the product AB is 6 times 8, what is the size of B?
9x8
Triangular matrix
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 n times n matrix has n pivot positions
Is it possible for a 5x5 atrix to be invertible when its columns do not span set of real numbers R^5? Why or why not?
B. It is not possible; according to the Invertible Matrix Theorem an nxn matrix cannot be invertible when its columns do not span R^n
An elementary row operation on A does not change the determinant. Choose the correct answer below.
False
Determine whether the statement "detA^T=(-1)detA is true or false
False
Determine whether the statement "A row replacement operation on A does not change the eigenvalues" is true or false. Choose the correct answer below.
False
Determine whether the statement "If A is 3x3 with columns a1,a2,a3, then det A equals the volume of the parallelepiped determined by a1,a2,a3 is true or false?
False
If A is invertible, then elementary row operations that reduce A to the identity In also reduce A^-1 to In
False
If lambda+5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
False
The determinant of A is the product of the diagonal entries in A. Select the correct choice below and, if necessary, fill in the answer box to complete your choice
False
The eigenvalues of a matrix are on its main diagonal. Choose the correct answer below
False
To find the eigenvalues of A, reduce A to echelon form. Choose the correct answer below.
False
if Ax= lamdax for some scalar lamda then x is an eigenvector of A. Choose the correct answer below.
False
if Ax=lamdax for some vector x, then lambda is an eigenvalue of A. Choose the correct answer below.
False
if v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues. Choose the correct answer below.
False
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.
(det A)(det B)=det AB. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
True
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-cl)x=0 has a nontrivial solution
True
An eigenspace of A is a null space of a certain matrix. Choose the correct answer below.
True
A steady-state vector for a stochastic matrix is actually an eigenvector. Choose the correct answer below.
True
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. 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 can be row reduced to the identity matrix, then A must be invertible.
True
If A is invertible, then the inverse of A^-1 is A itself.
True
if A= [a b] [c d]and ad=bc, then A is not invertible.
True
A product of invertible nxn matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
False
