 5.3.1: In Exercises 16, determine if Cramers Rule can be applied to findth...
 5.3.2: In Exercises 16, determine if Cramers Rule can be applied to findth...
 5.3.3: In Exercises 16, determine if Cramers Rule can be applied to findth...
 5.3.4: In Exercises 16, determine if Cramers Rule can be applied to findth...
 5.3.5: In Exercises 16, determine if Cramers Rule can be applied to findth...
 5.3.6: In Exercises 16, determine if Cramers Rule can be applied to findth...
 5.3.7: In Exercises 712, find the value of x2 in the unique solution ofthe...
 5.3.8: In Exercises 712, find the value of x2 in the unique solution ofthe...
 5.3.9: In Exercises 712, find the value of x2 in the unique solution ofthe...
 5.3.10: In Exercises 712, find the value of x2 in the unique solution ofthe...
 5.3.11: In Exercises 712, find the value of x2 in the unique solution ofthe...
 5.3.12: In Exercises 712, find the value of x2 in the unique solution ofthe...
 5.3.13: In Exercises 1318, for the given matrix A, find adj(A) and thenuse ...
 5.3.14: In Exercises 1318, for the given matrix A, find adj(A) and thenuse ...
 5.3.15: In Exercises 1318, for the given matrix A, find adj(A) and thenuse ...
 5.3.16: In Exercises 1318, for the given matrix A, find adj(A) and thenuse ...
 5.3.17: In Exercises 1318, for the given matrix A, find adj(A) and thenuse ...
 5.3.18: In Exercises 1318, for the given matrix A, find adj(A) and thenuse ...
 5.3.19: In Exercises 1922, sketch the parallelogram with the given vertices...
 5.3.20: In Exercises 1922, sketch the parallelogram with the given vertices...
 5.3.21: In Exercises 1922, sketch the parallelogram with the given vertices...
 5.3.22: In Exercises 1922, sketch the parallelogram with the given vertices...
 5.3.23: In Exercises 2328, find the area of T(D) for T(x) = Ax.D is the rec...
 5.3.24: In Exercises 2328, find the area of T(D) for T(x) = Ax.D is the rec...
 5.3.25: In Exercises 2328, find the area of T(D) for T(x) = Ax.D is the par...
 5.3.26: In Exercises 2328, find the area of T(D) for T(x) = Ax.D is the par...
 5.3.27: In Exercises 2328, find the area of T(D) for T(x) = Ax.D is the par...
 5.3.28: In Exercises 2328, find the area of T(D) for T(x) = Ax.D is the par...
 5.3.29: In Exercises 2932, find a linear transformation T that gives aonet...
 5.3.30: In Exercises 2932, find a linear transformation T that gives aonet...
 5.3.31: In Exercises 2932, find a linear transformation T that gives aonet...
 5.3.32: In Exercises 2932, find a linear transformation T that gives aonet...
 5.3.33: For Exercises 3336: In three dimensions, Theorem 5.21 states thatif...
 5.3.34: For Exercises 3336: In three dimensions, Theorem 5.21 states thatif...
 5.3.35: For Exercises 3336: In three dimensions, Theorem 5.21 states thatif...
 5.3.36: For Exercises 3336: In three dimensions, Theorem 5.21 states thatif...
 5.3.37: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 5.3.38: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 5.3.39: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 5.3.40: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 5.3.41: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 5.3.42: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 5.3.43: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 5.3.44: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 5.3.45: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 5.3.46: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 5.3.47: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 5.3.48: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 5.3.49: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 5.3.50: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 5.3.51: Prove that the linear transformation T(x) = Bx withB =b1 00 b2gives...
 5.3.52: Prove that the linear transformation T(x) = Ax withA =a 00 bgives a...
 5.3.53: Suppose that A is a 33 matrix with det(A) = 2. Show thatA adj(A) is...
 5.3.54: Prove that if A is an n n matrix with linearly independentcolumns, ...
 5.3.55: Show that if A is an n n symmetric matrix, then adj(A) isalso symme...
 5.3.56: Prove that if A is an n n diagonal matrix, then so is adj(A).
 5.3.57: Suppose that A is an n n matrix and c is a scalar. Prove thatadj(c ...
 5.3.58: Prove that if Ais an invertible nn matrix, then det(adj(A)) = det(A...
 5.3.59: Suppose that A is an invertible n n matrix. Show thatadj(A)1 = adj(...
 5.3.60: Suppose that A is an invertible n n matrix and that both Aand A1 ha...
 5.3.61: Prove that if A is a diagonal matrix, then so is adj(A).
 5.3.62: In this problem, we show thatarea(T ) = 12  det(A),where area(T )...
 5.3.63: C In Exercises 6366, use Cramers Rule to find the solution tothe sy...
 5.3.64: C In Exercises 6366, use Cramers Rule to find the solution tothe sy...
 5.3.65: C In Exercises 6366, use Cramers Rule to find the solution tothe sy...
 5.3.66: C In Exercises 6366, use Cramers Rule to find the solution tothe sy...
 5.3.67: C In Exercises 6770, for the given matrix A, find adj(A) andthen us...
 5.3.68: C In Exercises 6770, for the given matrix A, find adj(A) andthen us...
 5.3.69: C In Exercises 6770, for the given matrix A, find adj(A) andthen us...
 5.3.70: C In Exercises 6770, for the given matrix A, find adj(A) andthen us...
Solutions for Chapter 5.3: Applications of the Determinant
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 5.3: Applications of the Determinant
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. Chapter 5.3: Applications of the Determinant includes 70 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 70 problems in chapter 5.3: Applications of the Determinant have been answered, more than 15064 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.