 3.1: True or Falsedel(A + 8) = del(A) + del(B)
 3.2: True or False del(A  IB) = ~ del(A)
 3.3: True or FalseIf del(A) = O. then A has alleasl lwoequal rows.
 3.4: True or FalseIf A a~ a column of all zeroS. then del(A} = O.
 3.5: True or FalseA is singular if and only if det(A) = O.
 3.6: True or False If 8 is the reduced row echelon fonn of A. then det(B...
 3.7: True or FalseThe determinant oran elementary millrix is always I.
 3.8: True or FalseIf A is I1Ollsingular. then A  I _ ~ ~IJj(A). det(A )
 3.9: True or FalseIf T i~ a matrix transformation frolll R.2 ..... R. 2 ...
 3.10: True or FalseIf aJllhe diagonal elements of an /I X /I m.1lrix A ar...
 3.11: True or False det(ABTA  J)= detB.
 3.12: True or False (del cA) = det(A).
Solutions for Chapter 3: Determinants
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780132296540
Solutions for Chapter 3: Determinants
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Chapter 3: Determinants includes 12 full stepbystep solutions. Since 12 problems in chapter 3: Determinants have been answered, more than 12523 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.