LinFinal
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
