Test 2 TF

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An m×n upper triangular matrix is one whose entries below the main diagonal are​ zeros, as is shown in the matrix to the right. When is a square upper triangular matrix​ invertible?

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×n matrix has n pivot positions.

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

A matrix A is diagonalizable if A has n eigenvectors

False

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

A subset H of set of real numbers ℝn is a subspace if the zero vector is in H.

False

A subspace of set of real numbers ℝn is any set H such that​ (i) the zero vector is in​ H, (ii) u​, v​, and u+v are in​ H, and​ (iii) c is a scalar and cu is in H.

False

An elementary row operation on A does not change the determinant.

False

Determine whether the statement "det(A^T) = (-1)det(A) is true or false

False

Determine whether the statement​ "A row replacement operation on A does not change the​ eigenvalues" is true or false

False

Determine whether the statement​ "If A is 3×​3, 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​ 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 A is​ invertible, then elementary row operations that reduce A to the identity In also reduce A^−1 to In

False

If Ax = λx for some vector x​, then λ is an eigenvalue of A.

False

If B is an echelon form of a matrix​ A, then the pivot columns of B form a basis for Col(A)

False

If det(A) is​ zero, then two rows or two columns are the​ same, or a row or a column is zero

False

If three row interchanges are made in​ succession, then the new determinant equals the old determinant.

False

If v1 and v2 are linearly independent​ eigenvectors, then they correspond to distinct eigenvalues

False

If λ+5 is a factor of the characteristic polynomial of​ A, then 5 is an eigenvalue of A

False

The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row

False

The column space of a matrix A is the set of solutions of Ax = b.

False

The determinant of A is the product of the diagonal entries in A

False

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

The determinant of a triangular matrix is the sum of the entries on the main diagonal.

False

The eigenvalues of a matrix are on its main diagonal.

False

The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of set of real numbers ℝm.

False

To find the eigenvalues of​ A, reduce A to echelon form

False

det(A+​B) = det(A) + det(B)

False

det(A^-1) =​ (−​1)det(A)

False

Suppose F is a 5×5 matrix whose column space is not equal to set of real numbers ℝ5. What can you say about Nul​(F)?

If Col(F) ≠ set of real numbers ℝ5​, then the columns of F do not span set of real numbers ℝ5. Since F is​ square, the Invertible Matrix Theorem shows that F is not invertible and the equation Fx = 0 has a nontrivial solution.​ Therefore, Nul(F) contains a nonzero vector.

If Q is a 4x4 matrix and Col(Q) = set of real numbers ℝ4​, what can you say about solutions of equations of the form Qx = b for b in set of real numbers ℝ4​?

If Col(Q) = set of real numbers ℝ4​, then the columns of Q span set of real numbers ℝ4. Since Q is​ square, the Invertible Matrix Theorem shows that Q is invertible and the equation Qx = b has a solution for each b in set of real numbers ℝ4.

If R is a 6×6 matrix and Nul(R) is not the zero​ subspace, what can you say about Col​(R)?

If Nul(R) contains nonzero​ vectors, then the equation Rx = 0 has nontrivial solutions. Since R is​ square, the Invertible Matrix Theorem shows that R is not invertible and the columns of R do not span set of real numbers ℝ6. Therefore, Col(R) is a subspace of set of real numbers ℝ6​, but Col(R) not equals set of real numbers Col(R) ≠ ℝ6.

Suppose A is n×n and the equation Ax = b has a solution for each b in set of real numbers ℝn. Explain why A must be invertible.​

If the equation Ax = b has a solution for each b in set of real numbers ℝn​, then A has a pivot position in each row. Since A is​ square, the pivots must be on the diagonal of A. It follows that A is row equivalent to In. ​Therefore, A is invertible.

If A is​ invertible, then the columns of A^−1 are linearly independent.

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?

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

Is it possible for a 5x5 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.

How many rows does B have if BC is a 5×2 matrix?

Matrix B has 5 rows

A is a 3×3 matrix with two eigenvalues. Each eigenspace is​ one-dimensional. Is A​ diagonalizable?

No. The sum of the dimensions of the eigenspaces equals 2 and the matrix has 3 columns. The sum of the dimensions of the eigenspace and the number of columns must be equal.

Explain why the columns of an n×n matrix A span set of real numbers ℝn when A is invertible

Since A is​ invertible, for each b in set of real numbers ℝn the equation Ax = b has a unique solution. Since the equation Ax = b has a solution for all b in set of real numbers ℝn​, the columns of A span set of real numbers ℝn.

Suppose A is n×n and the equation Ax = 0 has only the trivial solution. Explain why A has n pivot columns and A is row equivalent to In.

Suppose A is n×n and the equation Ax = 0 has only the trivial solution. Then there are no free variables in this​ equation, thus A has n pivot columns. Since A is square and the n pivot positions must be in different​ rows, the pivots in an echelon form of A must be on the main diagonal. Hence A is row equivalent to the n×n identity​ matrix, In.

If the given equation Gx = y has more than one solution for some y in set of real numbers ℝn​, can the columns of G span set of real numbers ℝn​?

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

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 matrix A is 2×9 and the product AB is 2×7​, what is the size of​ B?

The size of B is 9x7

A matrix A is not invertible if and only if 0 is an eigenvalue of A

True

A number c is an eigenvalue of A if and only if the equation (A−​cI)x = 0 has a nontrivial solution

True

A row replacement operation does not affect the determinant of a matrix

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

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 eigen vector of A may be​ difficult, but checking whether a given vector u is in fact an eigen vector is easy

True

Given vectors v1 ,..., vp in set of real numbers ℝn​, the set of all linear combinations of these vectors is a subspace of set of real numbers ℝn.

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=Start 2 By 2 Table 1st Row 1st Column a 2nd Column b 2nd Row 1st Column c 2nd Column d EndTable abcd and ad=​bc, then A is not invertible

True

If AP = ​PD, with D​ diagonal, then the nonzero columns of P must be eigenvectors of A.

True

If Ax = λx for some scalar λ​, then x is an eigenvector of A

True

If set of real numbers ℝn has a basis of eigenvectors of​ A, then A is diagonalizable

True

If the columns of A are linearly​ dependent, then det(A) = 0.

True

If v1 ,..., vp are in ℝn​, then S = Span{v1 ,..., vp} is the same as the column space of the matrix A = [v1 ... vp].

True

Row operations do not affect linear dependence relations among the columns of a matrix.

True

The columns of an invertible n×n matrix form a basis for set of real numbers ℝn

True

The null space of an m×n matrix is a subspace of set of real numbers ℝn.

True

[det(A)][det(B)] = det(AB)

True

If the subspace of all solutions of Ax = 0 has a basis consisting of four vectors and if A is a 6x9 matrix, what is the rank of​ A?

rank A = 5

What is the rank of a 7x9 matrix whose null space is three dimensional?

rank A = 6

If the rank of a 4x7 matrix A is 22​, what is the dimension of the solution space Ax = 0​?

the dimension of the solution space is 5


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