LA TF 2

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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


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