Linear Algebra

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Inverse of a 2x2 matrix

(1/ad-bc)[d -b] [-c a]

If A is​ invertible, then elementary row operations that reduce A ^n also reduce A ^1 to Upper I^n

false

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

false

transpose of identity matrix is identity matrix

true

If A and B are nxn and invertible, then B^-1A^-1 is the inverse of AB

true (pay attention to order)

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, u has to be nonzero and there has to be a nontrivial solution

subspace of Rn

-zero vector in H -for each u and v in H the sum u+v is in H -for each u i H and each scalar c, the vector cu is in H

A is invertible if eigenvalue...

0 not eigenvalue of A

det(A^-1)

1/det(A)

If A is nxn invertible matrix then for each b in R^n the equation has a unique solution x=....

A^-1b

(A+B)^T

A^T+B^T

(AB^T)

B^TA^T

dimension of col A

aka rank number of pivots

what must be true for a square upper triangular matrix to be invertible

all diagonal entries must be nonzero

det(AB)

det(A)det(B)

det(A^T)

detA

. 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

A product of invertible n xn 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 R Superscript n is a subspace if the zero vector is in H.

false

A subspace of set of real numbers R Superscript n is any set H such that​ (i) the zero vector is in​ H, (ii) Bold u ​, Bold v ​, and Bold u plusBold v are in​ H, and​ (iii) c is a scalar and cBold u is in H.

false

AB=AC then B=C always

false

AB=BA

false

Each line in set of real numbers R Superscript n is a​ one-dimensional subspace of set of real numbers R Superscript n . 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

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

false

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

false

he dimension of Nul A is the number of variables in the equation Axequals 0. Choose the correct answer below.

false

​det(A+B)=det A+det B

false

If Axequals lambdax for some vector x​, then lambda is an eigenvalue of A. Choose the correct answer below.

false, must also have nontrvial solution

Can a square matrix with two identical rows be invertible?

no bc linearlly dependent

dimension of null A

number of free variables

(rA)^T

rA^T

(A^T)^T

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 ​(Aminus ​cI)xequals0 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

A​ steady-state vector for a stochastic matrix is actually an eigenvector. Choose the correct answer below.

true

Given vectors Bold v 1 ​, ​..., Bold v Subscript p in set of real numbers R Superscript n ​, the set of all linear combinations of these vectors is a subspace of set of real numbers R Superscript n .

true

If A is an nxn matric and if k is a positive integer, then A^k denotes the product of k copies of A

true

If A is invertible, then the inverse of A^-1 is A itself

true

If A is invertible, then the inverse of A^T is (A^-1)^T

true

If B equalsStartSet Bold v 1 comma ... comma Bold v Subscript p EndSet is a basis for a subspace H of set of real numbers R Superscript n ​, then the correspondence Bold x maps to left bracket Bold x right bracket Subscript Upper B makes H look and act the same as set of real numbers R Superscript p .

true

If Bequals StartSet Bold v 1 comma ... comma Bold v Subscript p EndSet is a basis for a subspace H and if xequals c 1 Bold v 1 plus ... plus c Subscript p Baseline Bold v Subscript p​, then c 1 ​,...,c Subscript p are the coordinates of x relative to the basis B.

true

If Bold v 1,..., v p are in set of real numbers R Superscript n​, then Sequals Span StartSet Bold v 1 comma ... comma Bold v Subscript p Baseline EndSet is the same as the column space of the matrix Aequals left bracket Bold v 1 font size decreased by 3 times times times Bold v Subscript p Baseline right bracket.

true

If H is a​ p-dimensional subspace of set of real numbers R Superscript n ​, then a linearly independent set of p vectors in H is a basis for H. Choose the correct answer below.

true

If Upper B is a basis for a subspace​ H, then each vector in H can be written in only one way as a linear combination of the vectors in B .

true

If a set of p vectors spans a​ p-dimensional subspace H of set of real numbers R Superscript n ​, then these vectors form a basis of H. Choose the correct answer below.

true

If k=0 then A^k is identified with identity matrix

true

any matrix times I is the same matrix

true


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