 3.1.1E: Compute the determinants in Exercises 1–8 using a cofactor expansio...
 3.1.2E: Compute the determinants in Exercises 1–8 using a cofactor expansio...
 3.1.3E: Compute the determinants in Exercises 1–8 using a cofactor expansio...
 3.1.4E: Compute the determinants in Exercises 1–8 using a cofactor expansio...
 3.1.5E: Compute the determinants in Exercises 1–8 using a cofactor expansio...
 3.1.6E: Compute the determinants in Exercises 1–8 using a cofactor expansio...
 3.1.7E: Compute the determinants in Exercises 1–8 using a cofactor expansio...
 3.1.8E: Compute the determinants in Exercises 1–8 using a cofactor expansio...
 3.1.9E: Compute the determinants in Exercises 9–14 by cofactor expansions. ...
 3.1.10E: Compute the determinants in Exercises 9–14 by cofactor expansions. ...
 3.1.11E: Compute the determinants in Exercises 9–14 by cofactor expansions. ...
 3.1.12E: Compute the determinants in Exercises 9–14 by cofactor expansions. ...
 3.1.13E: Compute the determinants in Exercises 9–14 by cofactor expansions. ...
 3.1.14E: Compute the determinants in Exercises 9–14 by cofactor expansions. ...
 3.1.15E: The expansion of a 3 × 3 determinant can be remembered by the follo...
 3.1.16E: The expansion of a 3 × 3 determinant can be remembered by the follo...
 3.1.17E: The expansion of a 3 × 3 determinant can be remembered by the follo...
 3.1.18E: The expansion of a 3 × 3 determinant can be remembered by the follo...
 3.1.19E: In Exercises 19–24, explore the effect of an elementary row operati...
 3.1.20E: In Exercises 19–24, explore the effect of an elementary row operati...
 3.1.21E: In Exercises 19–24, explore the effect of an elementary row operati...
 3.1.22E: In Exercises 19–24, explore the effect of an elementary row operati...
 3.1.23E: In Exercises 19–24, explore the effect of an elementary row operati...
 3.1.24E: In Exercises 19–24, explore the effect of an elementary row operati...
 3.1.25E: Compute the determinants of the elementary matrices given in Exerci...
 3.1.26E: Compute the determinants of the elementary matrices given in Exerci...
 3.1.27E: Compute the determinants of the elementary matrices given in Exerci...
 3.1.28E: Compute the determinants of the elementary matrices given in Exerci...
 3.1.29E: Compute the determinants of the elementary matrices given in Exerci...
 3.1.30E: Compute the determinants of the elementary matrices given in Exerci...
 3.1.31E: Use Exercises 25–28 to answer the questions in Exercises 31 and 32....
 3.1.32E: Use Exercises 25–28 to answer the questions in Exercises 31 and 32....
 3.1.33E: In Exercises 33–36, verify that det EA = (det E) (det A), where E i...
 3.1.34E: In Exercises 33–36, verify that det EA = (det E) (det A), where E i...
 3.1.35E: In Exercises 33–36, verify that det EA = (det E) (det A), where E i...
 3.1.36E: In Exercises 33–36, verify that det EA = (det E) (det A), where E i...
 3.1.37E: Let Write 5A. Is det 5A = 5 det A?
 3.1.38E: Let and let k be a scalar. Find a formula that relates det kA to k ...
 3.1.39E: a. An n × n determinant is defined by determinants of submatrices.b...
 3.1.40E: a. The cofactor expansion of det A down a column is the negative of...
 3.1.41E: Compute the area of the parallelogram determined by u, v, u + v, an...
 3.1.42E: where a, b, c are positive (for simplicity). Compute the area of th...
 3.1.43E: [M] Is it true that det. To find out, generate random 5 × 5 matrice...
 3.1.44E: [M] Is it true that det AB = (det A)(det B)? Experiment with four p...
 3.1.45E: [M] Construct a random 4 × 4 matrix A with integer entries between ...
 3.1.46E: [M] How is det A–1 related to det A? Experiment with random n × n i...
Solutions for Chapter 3.1: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 3.1
Get Full SolutionsChapter 3.1 includes 46 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. Since 46 problems in chapter 3.1 have been answered, more than 35380 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).