LinFinal

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When A is invertible, the det A = _____; when A is not invertible, the det A = ___

(-1)^r * product of pivots in U, where r is the number of row interchanges 0

Chapter 5; Thm 3 Properties of determinants det AB =

(det A)(det B)

Ch 3; Thm 3 If two rows of A are interchanged to produce B, then det B =

- det A

Steps to diagonalize a matrix

1. find eigenvalues of A 2. find eigenvectors of A 3. construct P from the vectors in step 2 4. construct D from the corresponding P; that is, REF of P

A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix, that is, if ....

A = P*D*P^-1 for some invertible matrix P and some diagonal matrix D

Definition of an eigenvector

An eigenvector of an n x n matrix A is a nonzero vector x such that Ax = lambda*x for some scalar lambda. A scalar lambda is called an eigenvalue of A if there is a nontrivial solution x of Ax = lambda*x; such an x is called an eigenvector corresponding to lambda

T/F det A^T = -det A

Fale

An elementary row operation on A does not change the determinant

False

If A is a 3 x 3 with columns a_1, a_2, a_3, then det A equals the volume of the parallelpiped determined by a1, a2, a3

False

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

False

T/F If A and B are n x n matrices, with det A = 2 and det B = 3, then det (A + B) = 5

False

T/F If A is diagonalizable, then A is invertible

False

T/F If A is invertible, then det A^-1 = det A

False

T/F If v_1 and v_2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues

False

T/F The determinant of A is the product of the diagonal entries in A

False

T/F det (A + B) = det A + det B

False

T/F if B is produced by interchanging two rows of A, then det B = det A

False

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

False

T/F To find the eigenvalues of A, reduce A to echelon form

False Row reduction can be used to find eigenvectors, but not eigen values

T/F det (-A) = - det A

False This statement is false for n x n invertible matrices with n an even integer

T/F det A^T = (-1) det A

False det (A^T) = det A when A is n x n

T/F If A is n x n and det A = 2, then det A^3 = 6

False, det A^3 = 2^3

T/F A is diagonalizable if and only if A has n eigenvalues, counting multiples

False.

T/F If Ax = lambda*x for some vector x, then lambda is an eigenvalue of A

False. The equation Ax = lambda*x must have a nontrivial soln

T/F A is diagonalizeable if A has n eigenvectors

False. The n eigenvectors must be linearly independent.

T/F det(A + B) = det A + det B

False; by the multiplicative property thm, det AB = (det A)(det B). This is not analogous for sums of matrices, in general

T/F Row reduction can be used to find eigenvalues

False; can only be used to find eigenvectors

T/F If A is a 3 x 3 matrix, then det 5 A = 5 det A

False; in this case, det 5 A = 5^3 det A

True/False The determinant of a triangular matrix is the sum of the entries on the main diagonal

False; it is the product of the diagonal entries

True/False The (i, j) cofactor of a matrix A is the matrix A_ij obtained by deleting from A its i-th row and j-th column

False; the cofactor is the determinant of this A_ij times -1^(i+j)

T/F If det A is zero, then two rows or two columns are the same, or a row or a column is zero

False; the converse is true, however

T/F A is diagonalizable if A = PDP^-1 for some matrix D and some invertible matrix P

False; the symbol D does not automatically denote a diagonal matrix

T/F If Ax = lambda*x for some scalar lambda, then x is an eigenvector of A

False; the vector x in Ax = lambda*x must be nonzero

T/F The eigenvalues of a matrix are on its main diagonal

False; triangular matrix

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

False; we can expand down any row or column and get the same determinant

Ch 3; Thm 6 Multiplicative property

If A and B are n x n matrices, then det AB = (det A)(det B)

If A and B are n x n matrices, then A is similar to B if there is an invertible matrix P such that...

P^-1*A*P = B, or equivalently, A = P*B*P^-1

(det A)(det B) = det AB

True

T/F A matrix A is not invertible IFF 0 is an eigen value of A

True

T/F A number c is an eigenvalue of A if and only if the equation (A - c*I)x = 0 has a nontrivial soln

True

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

True

T/F A steady-state vector for a stochastic matrix is actually an eigenvector

True

T/F Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy

True

T/F If A is invertible, then (det A)(det A^-1) = 1

True

T/F If B is formed by adding to one row of A a linear combination of the other rows, then det B = det A

True

T/F If B is produced by multiplying row 3 of A by 5, then det B = 5 * det A

True

T/F If R^n has a basis of eigenvectors of A, then A is diagonalizable

True

T/F If the columns of A are linearly dependent, then det A = 0

True

T/F If two row interchanges are made in succession, then the new determinant equals the old determinant

True

T/F If two rows of a 3 x 3 matrix A are the same, then det A = 0

True

T/F Row reduction can be used to find eigenvectors

True

T/F 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

True

True/False An n x n determinant is defined by determinants of (n - 1) x (n - 1) submatrices

True

T/F If A^3 = 0, then det A = 0

True det A^3 = (det A)^3

T/F det A^T*A is greater than or equal to zero

True det A^T * A = (det A)^2

T/F We can perform operations on the columns of a matrix in a way that is analogous to row operations

True. By Thm 5 If A is an n x n matrix, then det A^T = det A

T/F If A is a 2 x 2 matrix with a zero determinant, then one column of A is a multiple of the other

True. The columns of A are linearly dependent

T/F An eigenspace of A is a null space of a certain matrix

True; the eigenspace of A corresponding to lambda is the null space of the matrix A - lambda*I

Chapter 5; Thm 3 Properties of determinants A row scaling ...

also scales the determinant by the same scalar factor

When two columns or two rows of a matrix are the same, they are linearly (dependent/independent)

dependant

When a column or a row is zero, the columns/rows are (dependent/independent)

dependent

When the det A = 0, the columns of A or linearly (dependent/independent)

dependent

When the det A = 0, the rows of A or linearly (dependent/independent)

dependent

When the det A = 0, the rows of A or linearly (dependent/independent)

dependent *Rows of A are columns of A^T, and linearly dependent columns of A^T make A^T singular. When A^T is singular, so is A, by the invertible matrix thm.

A scalar lambda is an eigenvalue of an n x n matrix A IFF lambda satisfies the characteristic equation

det (A - lambda*I)=0

Let A and B be 4 x 4 matrices, with det A = -1 and det B = 2. Compute: det 2 A

det 2 A = 2^4 det A = 16 * -1 = -16

Let A and B be 3 x 3 matrices with det A = 4 and det B = -3. Computer det 5 A

det 5 A = 5^3 det A = 125 x 4 = 500

Ch 3; Thm 3 If a multiple of one row of A is added to another row to produce a matrix B, then det B =

det A

Ch 3; Thm 5 If A is an n x n matrix, then det A^T =

det A

Chapter 5; Thm 3 Properties of determinants det A^T =

det A

Suppose a square matrix A has been reduced to an echelon form U by row replacements and row interchanges. If there are r interchanges, then Thm 3 shows that det A =

det A = (-1)^r det U

Ch 3; Thm 1 The determinant of an n x n matrix A can be computed by a cofactor expansion across any row or down any column. The expansion across the ith row is.... The cofactor expansion down the j-th column is....

det A = a_i1*C_i1+a_i2*C_i2 + ... + a_in*C_in det A = a_1j*C_1j + a_2j*C_2j + ... + a_nj*C_nj

Let A and B be 4 x 4 matrices, with det A = -1 and det B = 2. Compute: det AB

det AB = (det A)(det B) = -1 * 2 = -2

Let A and B be 3 x 3 matrices with det A = 4 and det B = -3. Computer det AB

det AB = (det A)(det B) = 4 * -3 = -12

Let A and B be 3 x 3 matrices with det A = 4 and det B = -3. Computer det A^-1

det A^-1 = 1/det A = 1/4

Let A and B be 3 x 3 matrices with det A = 4 and det B = -3. Computer det A^3

det A^3 = (det A)^3 = 4^3 = 64

Let A and B be 4 x 4 matrices, with det A = -1 and det B = 2. Compute: det A^T * A

det A^T * A = (det A^T)(det A) = (det A)(det A) = -1 * -1 = 1

Let A and B be 4 x 4 matrices, with det A = -1 and det B = 2. Compute: det B^-1*A*B

det B^-1*A*B = (det B^-1)(det A)(det B) = (1/det B)(det A)(det B) = det A = -1

Let A and B be 4 x 4 matrices, with det A = -1 and det B = 2. Compute: det B^5

det B^5 = (det B)^5 = 2^5 = 32

Let A and B be 3 x 3 matrices with det A = 4 and det B = -3. Computer det B^T

det B^T = det B = -3

Chapter 5; Thm 3 Properties of determinants A row replacement operation on A (does/does not) change the determinant

does not

The set of all solutions of (A-lambda*I)x = 0 is just the null space of the matrix A - lambda*I. So this set is a subspace of R^n and is called the ____ of A corresponding to lambda. The eigenspace consists of _____

eigenspace the zero vector and all the eigenvectors corresponding to lambda

Ch 5; thm 4 If n x n matrices A and B are similar, then they have the same characteristic polynomial and hence the same...

eigenvalues (with the same multiplicities)

Ch 5; Thm 1 The eigenvalues of a triangular matrix are the...

entries on its main diagonal

A row replacement operation on A does not change the eigenvalues

false

det A^T = (-1)det A

false

Ch 3; Thm 3 If one row of A is multiplied by k to produce B, then det B =

k * det A

Ch 5; Thm 2 If v_1, ... , v_r are eigenvectors that correspond to distinct eigenvalues lambda_1, ... , lambda_r of an n x n matrix A, then the set {v_1, ... , v_r} is...

linearly dependent

Ch 5; Thm 5 The diagonalization thm An n x n matrix A is diagonalizable IFF A has n ___________ A is diagnoalizable IFF there are enough eigenvectors to ....

linearly independent eigenvectors form a basis of R^n. We call such a basis an eigenvector basis of R^n

Ch 5; Thm 6 An n x n matrix with ..... is diagnolaizable

n distinct eigenvalues

A 2 x 2 matrix is invertible if and only if its determinant is ____

nonzero

Invertible matrix thm Let A be an n x n matrix. Then A is invertible IFF The number 0 is.....

not an eigenvalue of A

Ch 3; Thm 4 A square matrix A is invertible if and only if the det A is

not equal to zero

Chapter 5; Thm 3 Properties of determinants A is invertible IFF det A is

not equal to zero

Invertible matrix thm Let A be an n x n matrix. Then A is invertible IFF the determinant of A is ...

not zero

Chapter 5; Thm 3 Properties of determinants If A is triangular, then det A is...

the product of the entries on the main diagonal of A

Chapter 5; Thm 3 Properties of determinants A row interchange changes....

the sign of the determinant

Ch 3; Thm 2 If A is a triangular matrix...

then det A is the product of the entries on the main diagonal of A.

The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A

true


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